Finding Composite and Inverse Functions

Find and Evaluate Composite Functions

Before we introduce the functions, we need to look at another operation on functions called composition. In composition, the output of one function is the input of a second function. For functions \(f\) and \(g,\) the composition is written \(f∘g\) and is defined by \((f∘g)(x)=f(g(x)).\)

We read \(f(g(x))\) as \(\text{“}f\) of \(g\) of \(x\text{.”}\)

To do a composition, the output of the first function, \(g(x),\) becomes the input of the second function, f, and so we must be sure that it is part of the domain of f.

We have actually used composition without using the notation many times before. When we graphed quadratic functions using translations, we were composing functions. For example, if we first graphed \(g(x)={x}^{2}\) as a parabola and then shifted it down vertically four units, we were using the composition defined by \((f∘g)(x)=f(g(x))\) where \(f(x)=x-4.\)

The next example will demonstrate that \((f∘g)(x),\) \((g∘f)(x)\) and \((f\cdot g)(x)\) usually result in different outputs.

Example

Try it.

For functions \(f(x)=4x-5\) and \(g(x)=2x+3,\) find: ⓐ \((f∘g)(x),\) ⓑ \((g∘f)(x),\) and ⓒ \((f\cdot g)(x).\)

Solution

ⓐ

Use the definition of \((f∘g)(x).\)
Distribute.
Simplify.

ⓑ

Use the definition of \((f∘g)(x).\)
Distribute.
Simplify.

Notice the difference in the result in part ⓐ and part ⓑ.

ⓒ Notice that \((f\cdot g)(x)\) is different than \((f∘g)(x).\) In part ⓐ we did the composition of the functions. Now in part ⓒ we are not composing them, we are multiplying them.

\(\begin{array}{llllll}\text{Use the definition of}\ (f\cdot g)(x). & & & & & (f\cdot g)(x)=f(x)\cdot g(x) \\ \text{Substitute}\ f(x)=4x-5\ \text{and}\ g(x)=2x+3. & & & & & (f\cdot g)(x)=(4x-5)\cdot (2x+3) \\ \text{Multiply.} & & & & & (f\cdot g)(x)=8{x}^{2}+2x-15\end{array}\)

In the next example we will evaluate a composition for a specific value.

Example

Try it.

For functions \(f(x)={x}^{2}-4,\) and \(g(x)=3x+2,\) find: ⓐ \((f∘g)(-3),\) ⓑ \((g∘f)(-1),\) and ⓒ \((f∘f)(2).\)

Solution

ⓐ

Use the definition of \((f∘g)(-3).\)
Simplify.
Simplify.

ⓑ

Use the definition of \((g∘f)(-1).\)
Simplify.
Simplify.

ⓒ

Use the definition of \((f∘f)(2).\)
Simplify.
Simplify.

Condensed: the full section is in OpenStax Intermediate Algebra 2e.

Determine Whether a Function is One-to-One

When we first introduced functions, we said a function is a relation that assigns to each element in its domain exactly one element in the range. For each ordered pair in the relation, each x-value is matched with only one y-value.

We used the birthday example to help us understand the definition. Every person has a birthday, but no one has two birthdays and it is okay for two people to share a birthday. Since each person has exactly one birthday, that relation is a function.

A function is one-to-one if each value in the range has exactly one element in the domain. For each ordered pair in the function, each y-value is matched with only one x-value.

Our example of the birthday relation is not a one-to-one function. Two people can share the same birthday. The range value August 2 is the birthday of Liz and June, and so one range value has two domain values. Therefore, the function is not one-to-one.

Example

Try it.

For each set of ordered pairs, determine if it represents a function and, if so, if the function is one-to-one.

ⓐ \(\{(-3,27),(-2,8),(-1,1),(0,0),(1,1),(2,8),(3,27)\}\) and ⓑ \(\{(0,0),(1,1),(4,2),(9,3),(16,4)\}.\)

Solution

ⓐ
\(\begin{array}{lllll} & & & & \ \{(-3,27),(-2,8),(-1,1),(0,0),(1,1),(2,8),(3,27)\}\end{array}\)

Each x-value is matched with only one y-value. So this relation is a function.

But each y-value is not paired with only one x-value, \((-3,27)\) and \((3,27),\) for example. So this function is not one-to-one.

ⓑ
\(\begin{array}{lllll} & & & & \ \{(0,0),(1,1),(4,2),(9,3),(16,4)\}\end{array}\)

Each x-value is matched with only one y-value. So this relation is a function.

Since each y-value is paired with only one x-value, this function is one-to-one.

The vertical line is representing an x-value and we check that it intersects the graph in only one y-value. Then it is a function.

Condensed: the full section is in OpenStax Intermediate Algebra 2e.

Find the Inverse of a Function

Let’s look at a one-to one function, \(f\), represented by the ordered pairs \(\{(0,5),(1,6),(2,7),(3,8)\}.\) For each \(x\)-value, \(f\) adds 5 to get the \(y\)-value. To ‘undo’ the addition of 5, we subtract 5 from each \(y\)-value and get back to the original \(x\)-value. We can call this “taking the inverse of \(f\)” and name the function \({f}^{-1}.\)

Notice that that the ordered pairs of \(f\) and \({f}^{-1}\) have their \(x\)-values and \(y\)-values reversed. The domain of \(f\) is the range of \({f}^{-1}\) and the domain of \({f}^{-1}\) is the range of \(f.\)

In the next example we will find the inverse of a function defined by ordered pairs.

Example

Try it.

Find the inverse of the function \(\{(0,3),(1,5),(2,7),(3,9)\}.\) Determine the domain and range of the inverse function.

Solution

This function is one-to-one since every \(x\)-value is paired with exactly one \(y\)-value.

To find the inverse we reverse the \(x\)-values and \(y\)-values in the ordered pairs of the function.

Function	\(\{(0,3),(1,5),(2,7),(3,9)\}\)
Inverse Function	\(\{(3,0),(5,1),(7,2),(9,3)\}\)
Domain of Inverse Function	\(\{3,5,7,9\}\)
Range of Inverse Function	\(\{0,1,2,3\}\)

We just noted that if \(f(x)\) is a one-to-one function whose ordered pairs are of the form \((x,y),\) then its inverse function \({f}^{-1}(x)\) is the set of ordered pairs \((y,x).\)

So if a point \((a,b)\) is on the graph of a function \(f(x),\) then the ordered pair \((b,a)\) is on the graph of \({f}^{-1}(x).\)

The distance between any two pairs \((a,b)\) and \((b,a)\) is cut in half by the line \(y=x.\) So we say the points are mirror images of each other through the line \(y=x.\)

Since every point on the graph of a function \(f(x)\) is a mirror image of a point on the graph of \({f}^{-1}(x),\) we say the graphs are mirror images of each other through the line \(y=x.\) We will use this concept to graph the inverse of a function in the next example.

Condensed: the full section is in OpenStax Intermediate Algebra 2e.

Key Concepts

• Composition of Functions: The composition of functions \(f\) and \(g,\) is written \(f∘g\) and is defined by
  \[(f∘g)(x)=f(g(x))\]
  We read \(f(g(x))\) as \(f\) of \(g\) of \(x.\)
• Horizontal Line Test: If every horizontal line, intersects the graph of a function in at most one point, it is a one-to-one function.
• Inverse of a Function Defined by Ordered Pairs: If \(f(x)\) is a one-to-one function whose ordered pairs are of the form \((x,y),\) then its inverse function \({f}^{-1}(x)\) is the set of ordered pairs \((y,x).\)
• Inverse Functions: For every \(x\) in the domain of one-to-one function \(f\) and \({f}^{-1},\)
  \[\begin{array}{lll}{f}^{-1}(f(x)) & = & x \\ f({f}^{-1}(x)) & = & x\end{array}\]
• How to Find the Inverse of a One-to-One Function:
  1. Substitute y for \(f(x).\)
  2. Interchange the variables x and y.
  3. Solve for y.
  4. Substitute \({f}^{-1}(x)\) for \(y.\)
  5. Verify that the functions are inverses.

Finding Composite and Inverse Functions

Find and Evaluate Composite Functions

In the following exercises, find ⓐ (f ∘ g)(x), ⓑ (g ∘ f)(x), and ⓒ (f · g)(x).

Try it.

\(f(x)=4x+3\) and \(g(x)=2x+5\)

Solution

ⓐ \(8x+23\) ⓑ \(8x+11\) ⓒ
\(8{x}^{2}+26x+15\)

Try it.

\(f(x)=3x-1\) and \(g(x)=5x-3\)

Try it.

\(f(x)=6x-5\) and \(g(x)=4x+1\)

Solution

ⓐ \(24x+1\) ⓑ \(24x-19\)
ⓒ \(24{x}^{2}-14x-5\)

Try it.

\(f(x)=2x+7\) and \(g(x)=3x-4\)

Try it.

\(f(x)=3x\) and \(g(x)=2{x}^{2}-3x\)

Solution

ⓐ \(6{x}^{2}-9x\) ⓑ \(18{x}^{2}-9x\)
ⓒ \(6{x}^{3}-9{x}^{2}\)

Try it.

\(f(x)=2x\) and \(g(x)=3{x}^{2}-1\)

Try it.

\(f(x)=2x-1\) and \(g(x)={x}^{2}+2\)

Solution

ⓐ \(2{x}^{2}+3\) ⓑ \(4{x}^{2}-4x+3\)
ⓒ \(2{x}^{3}-{x}^{2}+4x-2\)

Try it.

\(f(x)=4x+3\) and \(g(x)={x}^{2}-4\)

In the following exercises, find the values described.

Try it.

For functions \(f(x)=2{x}^{2}+3\) and \(g(x)=5x-1,\) find ⓐ \((f∘g)(-2)\) ⓑ \((g∘f)(-3)\) ⓒ \((f∘f)(-1)\)

Solution

ⓐ 245 ⓑ 104 ⓒ 53

Try it.

For functions \(f(x)=5{x}^{2}-1\) and \(g(x)=4x-1,\) find ⓐ \((f∘g)(1)\) ⓑ \((g∘f)(-1)\) ⓒ \((f∘f)(2)\)

Try it.

For functions \(f(x)=2{x}^{3}\) and \(g(x)=3{x}^{2}+2,\) find ⓐ \((f∘g)(-1)\) ⓑ \((g∘f)(1)\) ⓒ \((g∘g)(1)\)

Solution

ⓐ 250 ⓑ 14 ⓒ 77

Try it.

For functions \(f(x)=3{x}^{3}+1\) and \(g(x)=2{x}^{2}-3,\) find ⓐ \((f∘g)(-2)\) ⓑ \((g∘f)(-1)\) ⓒ \((g∘g)(1)\)

Determine Whether a Function is One-to-One

In the following exercises, determine if the set of ordered pairs represents a function and if so, is the function one-to-one.

Try it.

\(\{(-3,9),(-2,4),(-1,1),(0,0)\),
\((1,1),(2,4),(3,9)\)

Solution

Function; not one-to-one

Try it.

\(\{(9,-3),(4,-2),(1,-1),(0,0)\),
\((1,1),(4,2),(9,3)\)

Try it.

\(\{(-3,-5),(-2,-3),(-1,-1)\),
\((0,1),(1,3),(2,5),(3,7)\)

Solution

One-to-one function

Try it.

\(\{(5,3),(4,2),(3,1),(2,0)\),
\((1,-1),(0,-2),(-1,-3)\)

In the following exercises, determine whether each graph is the graph of a function and if so, is it one-to-one.

In the following exercises, find the inverse of each function. Determine the domain and range of the inverse function.

Try it.

\(\{(2,1),(4,2),(6,3),(8,4)\}\)

Solution

Inverse function: \(\{(1,2),(2,4),(3,6),(4,8)\}.\) Domain: \(\{1,2,3,4\}.\) Range: \(\{2,4,6,8\}.\)

Try it.

\(\{(6,2),(9,5),(12,8),(15,11)\}\)

Try it.

\(\{(0,-2),(1,3),(2,7),(3,12)\}\)

Solution

Inverse function: \(\{(-2,0),(3,1),(7,2),(12,3)\}.\) Domain: \(\{-2,3,7,12\}.\) Range: \(\{0,1,2,3\}.\)

Try it.

\(\{(0,0),(1,1),(2,4),(3,9)\}\)

Try it.

\(\{(-2,-3),(-1,-1),(0,1),(1,3)\}\)

Solution

Inverse function: \(\{(-3,\text{-}2),(-1,-1),(1,0),(3,1)\}.\) Domain: \(\{-3,\text{-}1,1,3\}.\) Range: \(\{-2,-1,0,1\}.\)

Try it.

\(\{(5,3),(4,2),(3,1),(2,0)\}\)

In the following exercises, graph, on the same coordinate system, the inverse of the one-to-one function shown.

In the following exercises, determine whether or not the given functions are inverses.

Try it.

\(f(x)=x+8\) and \(g(x)=x-8\)

Solution

\(g(f(x))=x,\) and \(f(g(x))=x,\) so they are inverses.

Try it.

\(f(x)=x-9\) and \(g(x)=x+9\)

Try it.

\(f(x)=7x\) and \(g(x)=\frac{x}{7}\)

Solution

\(g(f(x))=x,\) and \(f(g(x))=x,\) so they are inverses.

Try it.

\(f(x)=\frac{x}{11}\) and \(g(x)=11x\)

Try it.

\(f(x)=7x+3\) and \(g(x)=\frac{x-3}{7}\)

Solution

\(g(f(x))=x,\) and \(f(g(x))=x,\) so they are inverses.

Try it.

\(f(x)=5x-4\) and \(g(x)=\frac{x-4}{5}\)

Try it.

\(f(x)=\sqrt{x+2}\) and \(g(x)={x}^{2}-2(x>0)\)

Solution

\(g(f(x))=x,\) and \(f(g(x))=x,\) so they are inverses (for nonnegative \(x).\)

Try it.

\(f(x)=\sqrt[3]{x-4}\) and \(g(x)={x}^{3}+4\)

In the following exercises, find the inverse of each function.

Try it.

\(f(x)=x-12\)

Solution

\({f}^{-1}(x)=x+12\)

Try it.

\(f(x)=x+17\)

Try it.

\(f(x)=\frac{x}{6}\)

Solution

\({f}^{-1}(x)=6x\)

Try it.

\(f(x)=\frac{x}{4}\)

Try it.

\(f(x)=6x-7\)

Solution

\({f}^{-1}(x)=\frac{x+7}{6}\)

Try it.

\(f(x)=7x-1\)

Try it.

\(f(x)=-2x+5\)

Solution

\({f}^{-1}(x)=\frac{x-5}{-2}\)

Try it.

\(f(x)=-5x-4\)

Try it.

\(f(x)={x}^{2}+6,\) \(x\ge 0\)

Solution

\({f}^{-1}(x)=\sqrt{x-6}\)

Try it.

\(f(x)={x}^{2}-9,\) \(x\ge 0\)

Try it.

\(f(x)={x}^{3}-4\)

Solution

\({f}^{-1}(x)=\sqrt[3]{x+4}\)

Try it.

\(f(x)={x}^{3}+6\)

Try it.

\(f(x)=\frac{1}{x+2}\)

Solution

\({f}^{-1}(x)=\frac{1}{x}-2\)

Try it.

\(f(x)=\frac{1}{x-6}\)

Try it.

\(f(x)=\sqrt{x-2},\) \(x\ge 2\)

Solution

\({f}^{-1}(x)={x}^{2}+2\), \(x\ge 0\)

Try it.

\(f(x)=\sqrt{x+8},\) \(x\ge -8\)

Try it.

\(f(x)=\sqrt[3]{x-3}\)

Solution

\({f}^{-1}(x)={x}^{3}+3\)

Try it.

\(f(x)=\sqrt[3]{x+5}\)

Try it.

\(f(x)=\sqrt[4]{9x-5},\) \(x\ge \frac{5}{9}\)

Solution

\({f}^{-1}(x)=\frac{{x}^{4}+5}{9}\), \(x\ge 0\)

Try it.

\(f(x)=\sqrt[4]{8x-3},\) \(x\ge \frac{3}{8}\)

Try it.

\(f(x)=\sqrt[5]{-3x+5}\)

Solution

\({f}^{-1}(x)=\frac{{x}^{5}-5}{-3}\)

Try it.

\(f(x)=\sqrt[5]{-4x-3}\)

Condensed: the full section is in OpenStax Intermediate Algebra 2e.

Tani ti. Asnjë kalkulator nuk e zgjidh këtë, por pjesët e saj janë të llogaritura. Provo një më poshtë, ose shkruaj tënde.

⌨ Lloji ✎ E vizatova.

Problemi yt matematikor.

π Zgjidh

Praktiko (40)

Provo secilin në letër së pari. Zbulo përgjigjen për të kontrolluar; ato të verifikuara mund të hapen në zgjidhës për çdo hap.

1. If \(f(x)=2x-3\) and \(g(x)={x}^{2}+2x-3,\) find \(f(4).\)

   Puno mbi të. Pjesmarrës

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   ⓐ \(f\left(4\right)=5\); ⓑ \(g\left(f\left(4\right)\right)=32\)

2. Solve for \(x,\) \(3x+2y=12.\)

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   \(x=-\frac{2}{3}y+4\)

3. Simplify: \(5\frac{(x+4)}{5}-4.\)

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   \(x\)

4. For functions \(f(x)=4x-5\) and \(g(x)=2x+3,\) find: ⓐ \((f∘g)(x),\) ⓑ \((g∘f)(x),\) and ⓒ \((f\cdot g)(x).\)

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   ⓐ
   

   Use the definition of \((f∘g)(x).\)
   Distribute.
   Simplify.

   
   
   ⓑ
   

   Use the definition of \((f∘g)(x).\)
   Distribute.
   Simplify.

   Notice the difference in the result in part ⓐ and part ⓑ.

   ⓒ Notice that \((f\cdot g)(x)\) is different than \((f∘g)(x).\) In part ⓐ we did the composition of the functions. Now in part ⓒ we are not composing them, we are multiplying them.
   
   \(\begin{array}{llllll}\text{Use the definition of}\ (f\cdot g)(x). & & & & & (f\cdot g)(x)=f(x)\cdot g(x) \\ \text{Substitute}\ f(x)=4x-5\ \text{and}\ g(x)=2x+3. & & & & & (f\cdot g)(x)=(4x-5)\cdot (2x+3) \\ \text{Multiply.} & & & & & (f\cdot g)(x)=8{x}^{2}+2x-15\end{array}\)

5. For functions \(f(x)=3x-2\) and \(g(x)=5x+1,\) find ⓐ \((f∘g)(x)\) ⓑ \((g∘f)(x)\) ⓒ \((f\cdot g)(x)\).

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   ⓐ \(15x+1\) ⓑ \(15x-9\)
   ⓒ \(15{x}^{2}-7x-2\)

6. For functions \(f(x)=4x-3,\) and \(g(x)=6x-5,\) find ⓐ \((f∘g)(x),\) ⓑ \((g∘f)(x),\) and ⓒ \((f\cdot g)(x).\)

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   ⓐ \(24x-23\) ⓑ \(24x-23\)
   ⓒ \(24{x}^{2}-38x+15\)

7. For functions \(f(x)={x}^{2}-4,\) and \(g(x)=3x+2,\) find: ⓐ \((f∘g)(-3),\) ⓑ \((g∘f)(-1),\) and ⓒ \((f∘f)(2).\)

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   ⓐ
   

   Use the definition of \((f∘g)(-3).\)
   Simplify.
   Simplify.

   
   
   ⓑ
   

   Use the definition of \((g∘f)(-1).\)
   Simplify.
   Simplify.

   
   
   ⓒ
   

   Use the definition of \((f∘f)(2).\)
   Simplify.
   Simplify.

8. For functions \(f(x)={x}^{2}-9,\) and \(g(x)=2x+5,\) find ⓐ \((f∘g)(-2),\) ⓑ \((g∘f)(-3),\) and ⓒ \((f∘f)(4).\)

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   ⓐ –8 ⓑ 5 ⓒ 40

9. For functions \(f(x)={x}^{2}+1,\) and \(g(x)=3x-5,\) find ⓐ \((f∘g)(-1),\) ⓑ \((g∘f)(2),\) and ⓒ \((f∘f)(-1).\)

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   ⓐ 65 ⓑ 10 ⓒ 5

10. For each set of ordered pairs, determine if it represents a function and, if so, if the function is one-to-one.

    ⓐ \(\{(-3,27),(-2,8),(-1,1),(0,0),(1,1),(2,8),(3,27)\}\) and ⓑ \(\{(0,0),(1,1),(4,2),(9,3),(16,4)\}.\)

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    ⓐ
    \(\begin{array}{lllll} & & & & \ \{(-3,27),(-2,8),(-1,1),(0,0),(1,1),(2,8),(3,27)\}\end{array}\)

    Each x-value is matched with only one y-value. So this relation is a function.

    But each y-value is not paired with only one x-value, \((-3,27)\) and \((3,27),\) for example. So this function is not one-to-one.

    ⓑ
    \(\begin{array}{lllll} & & & & \ \{(0,0),(1,1),(4,2),(9,3),(16,4)\}\end{array}\)

    Each x-value is matched with only one y-value. So this relation is a function.

    Since each y-value is paired with only one x-value, this function is one-to-one.

11. For each set of ordered pairs, determine if it represents a function and if so, is the function one-to-one.

    ⓐ \(\{(-3,-6),(-2,-4),(-1,-2),(0,0),(1,2),(2,4),(3,6)\}\) ⓑ \(\{(-4,8),(-2,4),(-1,2),(0,0),(1,2),(2,4),(4,8)\}\)

    Puno mbi të. Pjesmarrës

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    ⓐ One-to-one function
    ⓑ Function; not one-to-one

12. For each set of ordered pairs, determine if it represents a function and if so, is the function one-to-one.

    ⓐ \(\{(27,-3),(8,-2),(1,-1),(0,0),(1,1),(8,2),(27,3)\}\) ⓑ \(\{(7,-3),(-5,-4),(8,0),(0,0),(-6,4),(-2,2),(-1,3)\}\)

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    ⓐ Not a function
    ⓑ Function; not one-to-one

13. Determine ⓐ whether each graph is the graph of a function and, if so, ⓑ whether it is one-to-one.

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    ⓐ

    Since any vertical line intersects the graph in at most one point, the graph is the graph of a function. Since any horizontal line intersects the graph in at most one point, the graph is the graph of a one-to-one function.

    ⓑ

    Since any vertical line intersects the graph in at most one point, the graph is the graph of a function. The horizontal line shown on the graph intersects it in two points. This graph does not represent a one-to-one function.

14. Determine whether each graph is the graph of a function and, if so, whether it is one-to-one.

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    ⓐ Not a function ⓑ One-to-one function

15. Determine whether each graph is the graph of a function and, if so, whether it is one-to-one.

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    ⓐ Function; not one-to-one ⓑ One-to-one function

16. Find the inverse of the function \(\{(0,3),(1,5),(2,7),(3,9)\}.\) Determine the domain and range of the inverse function.

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    This function is one-to-one since every \(x\)-value is paired with exactly one \(y\)-value.

    To find the inverse we reverse the \(x\)-values and \(y\)-values in the ordered pairs of the function.
    

    Function	\(\{(0,3),(1,5),(2,7),(3,9)\}\)
    Inverse Function	\(\{(3,0),(5,1),(7,2),(9,3)\}\)
    Domain of Inverse Function	\(\{3,5,7,9\}\)
    Range of Inverse Function	\(\{0,1,2,3\}\)

17. Find the inverse of \(\{(0,4),(1,7),(2,10),(3,13)\}.\) Determine the domain and range of the inverse function.

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    Inverse function: \(\{(4,0),(7,1),(10,2),(13,3)\}.\) Domain: \(\{4,7,10,13\}.\) Range: \(\{0,1,2,3\}.\)

18. Find the inverse of \(\{(-1,4),(-2,1),(-3,0),(-4,2)\}.\) Determine the domain and range of the inverse function.

    Puno mbi të. Pjesmarrës

    Zbulo përgjigjen

    Inverse function: \(\{(4,-1),(1,-2),(0,-3),(2,-4)\}.\) Domain: \(\{0,1,2,4\}.\) Range: \(\{-4,-3,-2,-1\}.\)

19. Graph, on the same coordinate system, the inverse of the one-to one function shown.

    Puno mbi të. Pjesmarrës

    Zbulo përgjigjen

    We can use points on the graph to find points on the inverse graph. Some points on the graph are: \((-5,-3),(-3,-1),(-1,0),(0,2),(3,4)\).

    So, the inverse function will contain the points: \((-3,-5),(-1,-3),(0,-1),(2,0),(4,3)\).

    Notice how the graph of the original function and the graph of the inverse functions are mirror images through the line \(y=x.\)

20. Graph, on the same coordinate system, the inverse of the one-to one function.

    Puno mbi të. Pjesmarrës

21. Graph, on the same coordinate system, the inverse of the one-to one function.

    Puno mbi të. Pjesmarrës

22. Verify that \(f(x)=5x-1\) and \(g(x)=\frac{x+1}{5}\) are inverse functions.

    Puno mbi të. Pjesmarrës

    Zbulo përgjigjen

    The functions are inverses of each other if \(g(f(x))=x\) and \(f(g(x))=x.\)

    Substitute \(5x-1\) for \(f(x).\)
    Simplify.
    Simplify.
    Substitute \(\frac{x+1}{5}\) for \(g(x).\)
    Simplify.
    Simplify.

    Since both \(g(f(x))=x\) and \(f(g(x))=x\) are true, the functions \(f(x)=5x-1\) and \(g(x)=\frac{x+1}{5}\) are inverse functions. That is, they are inverses of each other.

23. Verify that the functions are inverse functions.

    \(f(x)=4x-3\) and \(g(x)=\frac{x+3}{4}.\)

    Puno mbi të. Pjesmarrës

    Zbulo përgjigjen

    \(g(f(x))=x,\) and \(f(g(x))=x,\) so they are inverses.

24. Verify that the functions are inverse functions.

    \(f(x)=2x+6\) and \(g(x)=\frac{x-6}{2}.\)

    Puno mbi të. Pjesmarrës

    Zbulo përgjigjen

    \(g(f(x))=x,\) and \(f(g(x))=x,\) so they are inverses.

25. Find the inverse of \(f(x)=4x+7.\)

    Puno mbi të. Pjesmarrës

26. Find the inverse of the function \(f(x)=5x-3.\)

    Puno mbi të. Pjesmarrës

    Zbulo përgjigjen

    \({f}^{-1}(x)=\frac{x+3}{5}\)

27. Find the inverse of the function \(f(x)=8x+5.\)

    Puno mbi të. Pjesmarrës

    Zbulo përgjigjen

    \({f}^{-1}(x)=\frac{x-5}{8}\)

28. Find the inverse of \(f(x)=\sqrt[5]{2x-3}.\)

    Puno mbi të. Pjesmarrës

    Zbulo përgjigjen

    	\(f(x)=\sqrt[5]{2x-3}\)
    Substitute \(y\) for \(f(x)\).	\(y=\sqrt[5]{2x-3}\)
    Interchange the variables \(x\) and \(y\).	\(x=\sqrt[5]{2y-3}\)
    Solve for \(y\).	\({(x)}^{5}={(\sqrt[5]{2y-3})}^{5}\)
    	\({x}^{5}=2y-3\)
    	\({x}^{5}+3=2y\)
    	\(\frac{{x}^{5}+3}{2}=y\)
    Substitute \({f}^{-1}(x)\) for \(y\).	\({f}^{-1}(x)=\frac{{x}^{5}+3}{2}\)
    Verify that the functions are inverses.
    \({f}^{-1}(f(x))\overset{?}{=}x\)	\(f({f}^{-1}(x))\overset{?}{=}x\)
    \({f}^{-1}(\sqrt[5]{2x-3})\overset{?}{=}x\)	\(f(\frac{{x}^{5}+3}{2})\overset{?}{=}x\)
    \(\frac{{(\sqrt[5]{2x-3})}^{5}+3}{2}\overset{?}{=}x\)	\(\sqrt[5]{2(\frac{{x}^{5}+3}{2})-3}\overset{?}{=}x\)
    \(\frac{2x-3+3}{2}\overset{?}{=}x\)	\(\sqrt[5]{{x}^{5}+3-3}\overset{?}{=}x\)
    \(\frac{2x}{2}\overset{?}{=}x\)	\(\sqrt[5]{{x}^{5}}\overset{?}{=}x\)
    \(x=x✓\)	\(x=x✓\)

29. Find the inverse of the function \(f(x)=\sqrt[5]{3x-2}.\)

    Puno mbi të. Pjesmarrës

    Zbulo përgjigjen

    \({f}^{-1}(x)=\frac{{x}^{5}+2}{3}\)

30. Find the inverse of the function \(f(x)=\sqrt[4]{6x-7}.\)

    Puno mbi të. Pjesmarrës

    Zbulo përgjigjen

    \({f}^{-1}(x)=\frac{{x}^{4}+7}{6}\)

31. \(f(x)=4x+3\) and \(g(x)=2x+5\)

    Puno mbi të. Pjesmarrës

    Zbulo përgjigjen

    ⓐ \(8x+23\) ⓑ \(8x+11\) ⓒ
    \(8{x}^{2}+26x+15\)

32. \(f(x)=3x-1\) and \(g(x)=5x-3\)

    Puno mbi të. Pjesmarrës

33. \(f(x)=6x-5\) and \(g(x)=4x+1\)

    Puno mbi të. Pjesmarrës

    Zbulo përgjigjen

    ⓐ \(24x+1\) ⓑ \(24x-19\)
    ⓒ \(24{x}^{2}-14x-5\)

34. \(f(x)=2x+7\) and \(g(x)=3x-4\)

    Puno mbi të. Pjesmarrës

35. \(f(x)=3x\) and \(g(x)=2{x}^{2}-3x\)

    Puno mbi të. Pjesmarrës

    Zbulo përgjigjen

    ⓐ \(6{x}^{2}-9x\) ⓑ \(18{x}^{2}-9x\)
    ⓒ \(6{x}^{3}-9{x}^{2}\)

36. \(f(x)=2x\) and \(g(x)=3{x}^{2}-1\)

    Puno mbi të. Pjesmarrës

37. \(f(x)=2x-1\) and \(g(x)={x}^{2}+2\)

    Puno mbi të. Pjesmarrës

    Zbulo përgjigjen

    ⓐ \(2{x}^{2}+3\) ⓑ \(4{x}^{2}-4x+3\)
    ⓒ \(2{x}^{3}-{x}^{2}+4x-2\)

38. \(f(x)=4x+3\) and \(g(x)={x}^{2}-4\)

    Puno mbi të. Pjesmarrës

39. For functions \(f(x)=2{x}^{2}+3\) and \(g(x)=5x-1,\) find ⓐ \((f∘g)(-2)\) ⓑ \((g∘f)(-3)\) ⓒ \((f∘f)(-1)\)

    Puno mbi të. Pjesmarrës

    Zbulo përgjigjen

    ⓐ 245 ⓑ 104 ⓒ 53

40. For functions \(f(x)=5{x}^{2}-1\) and \(g(x)=4x-1,\) find ⓐ \((f∘g)(1)\) ⓑ \((g∘f)(-1)\) ⓒ \((f∘f)(2)\)

    Puno mbi të. Pjesmarrës

Mbaje punën tënde

Një llogari e lirë shtohet shënime në çdo mësim, një regjistrim të asaj që ju keni përfunduar, problemet tuaja të zgjidhura në një vend, dhe një mësues që ju mund të pyesni rreth kësaj faqeje. Matematika vetë është e hapur për të gjithë, të regjistruar apo jo.

Regjistrohu Hyr

Simbolet e përdorura këtu

Prek çdo simbol për përkufizimin e plotë, një fotografi dhe se çfarë do të thotë çdo shkronjë në të.

\sqrt{x}square root, n-th root
The non-negative number whose square (n-th power) is x.\leqless/greater than or equal
Inequalities that allow equality; < and > exclude it.|x|absolute value / modulus
Distance from zero: |−3| = 3. For a complex number, distance from the origin.\binom{n}{k}binomial coefficient, "n choose k"
Number of k-element subsets of n things: n!/(k!(n−k)!).\pipi
Ratio of a circle's circumference to its diameter, 3.14159…eEuler's number
2.71828…, the base whose exponential is its own derivative.iimaginary unit
i² = −1.\thetatheta
The usual name for an angle.\log_b xlogarithm, natural log
The exponent b must be raised to for x; ln uses base e.\mathbf{v}vector
A quantity with magnitude and direction; a column of numbers.

Si: Finding Composite and Inverse Functions

1. Find and evaluate composite functions
2. Determine whether a function is one-to-one
3. Find the inverse of a function
4. Substitute
5. Interchange the variables
6. Solve for
7. Substitute
8. Verify that the functions are inverses.

Pyetja që bëjnë njerëzit

What is a function, really?

A rule that assigns exactly one output to each input. The vertical-line test on a graph is the same idea: no input may have two outputs.

Why do we need complex numbers?

Because x² + 1 = 0 has no real solution, and allowing one new number i with i² = −1 makes every polynomial equation solvable. They then turn out to describe rotation, waves and alternating current more naturally than real numbers do.

Pjesa e kësaj faqeje është adaptuar nga OpenStax Intermediate Algebra 2e (CC BY-NC-SA 4.0). E përmbledhur dhe ri-shkruar këtu; gabimet janë tona.

Më shumë në Precalculus

Complex numbersPolynomial functionsRational functionsSequences and seriesThe binomial theoremConic sectionsVectorsExponential and logarithmic functionsPolynomial division and the remainder theoremParametric equations and polar coordinates