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Chapter 9: Problem 31

Find the inverse of each one-to-one function. \(f(x)=5 x+2\)

Short Answer

Expert verified
The inverse function is \(f^{-1}(x) = \frac{x-2}{5}\).

Step by step solution

01

Understand the Function

The function given is a linear function of the form \(f(x) = 5x + 2\). To find the inverse, we need to express \(x\) in terms of \(f(x)\).
02

Replace f(x) with y

To simplify calculations, replace \(f(x)\) with \(y\), so we have the equation \(y = 5x + 2\).
03

Solve for x

Rearrange the equation to solve for \(x\). Subtract 2 from both sides to get \(y - 2 = 5x\). Then, divide both sides by 5 to find \(x\): \(x = \frac{y - 2}{5}\).
04

Express x in terms of f^{-1}(x)

The inverse function \(f^{-1}(x)\) is found by interchanging \(x\) and \(y\) in the equation from step 3. This gives us \(f^{-1}(x) = \frac{x - 2}{5}\).
05

Verify the Inverse

To confirm, compute \(f(f^{-1}(x))\) and \(f^{-1}(f(x))\) to ensure both equal \(x\). If substituting gives identity, then the inverse is correct.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Understanding Linear Functions
Linear functions are foundational in mathematics, especially in algebra. They produce straight-line graphs and follow the general form \( f(x) = mx + c \), where \( m \) is the slope and \( c \) is the y-intercept. Here, in our function \( f(x) = 5x + 2 \), \( m = 5 \) and \( c = 2 \).

This means the function grows by 5 units vertically for every increase of 1 unit horizontally. Linear functions are easy to work with because of their straightforward nature, making them an excellent starting point to learn about more complex mathematical concepts like inverse functions.
Algebraic Manipulation Techniques
Algebraic manipulation is a crucial skill for solving equations and finding inverses. It involves rearranging equations using addition, subtraction, multiplication, and division.

In the original solution, we start with the equation \( y = 5x + 2 \). To find the inverse, you rearrange it to solve for \( x \).
  • First, subtract 2 from both sides to isolate terms with \( x \): \( y - 2 = 5x \).
  • Next, divide every term by 5: \( x = \frac{y - 2}{5} \).
Algebraic manipulation like this helps in transforming one equation into another form, making it easier to solve or understand another property about the function.
Verifying Function Inverses
Verifying if a function and its potential inverse are truly inverses of each other is an essential step in mathematics. You need to check two compositions: \( f(f^{-1}(x)) \) and \( f^{-1}(f(x)) \). If both yield \( x \), then \( f(x) \) and \( f^{-1}(x) \) are indeed inverses.

For our function \( f(x) = 5x + 2 \) and its inverse \( f^{-1}(x) = \frac{x - 2}{5} \):
  • Compute \( f(f^{-1}(x)) = f\left(\frac{x - 2}{5}\right) = 5\left(\frac{x - 2}{5}\right) + 2 = x - 2 + 2 = x \).
  • Compute \( f^{-1}(f(x)) = \frac{5x + 2 - 2}{5} = x \).
Both compositions equal \( x \), confirming that \( f(x) \) and \( f^{-1} (x) \) are true inverses. This verification step ensures the mathematical correctness of our results.

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Most popular questions from this chapter

If \(x=-2, y=0,\) and \(z=3,\) find the value of each expression. See Section 1.3 $$ \frac{x^{3}-2 y+z}{2 z} $$

Graph each function and its inverse function on the same set of axes. Label any intercepts. $$ y=\left(\frac{1}{3}\right)^{x} ; y=\log _{1 / 3} x $$

Graph each function by finding ordered pair solutions, plotting the solutions, and then drawing a smooth curve through the plotted points. $$ f(x)=\ln x $$

Solve. See Example 4. $$ \log _{9} x=\frac{1}{2} $$

Add or subtract as indicated. See Section 6.2. $$ \frac{2}{x}+\frac{3}{x^{2}} $$
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