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Chapter 10: Problem 19

Determine whether each function is one-to-one. If it is, find the inverse. $$f(x)=2 x+4$$

Short Answer

Expert verified
The function \( f(x) = 2x + 4 \) is one-to-one. Its inverse is \( f^{-1}(x) = \frac{x-4}{2} \).

Step by step solution

01

Determine if the function is one-to-one

A function is one-to-one if every output is the result of exactly one input. To determine if the function is one-to-one, check if it passes the horizontal line test. For the function \( f(x) = 2x + 4 \), since it is a linear function with a non-zero slope (2), each output value will be produced by one unique input value. Hence, \( f(x) = 2x + 4 \) is one-to-one.
02

Find the inverse

To find the inverse of the function, start by replacing \( f(x) \) with \( y \). So, the function is \( y = 2x + 4 \). Now, solve for \( x \) in terms of \( y \). First, subtract 4 from both sides to get: \( y - 4 = 2x \) Next, divide both sides by 2: \( x = \frac{y-4}{2} \). Therefore, the inverse function is \( f^{-1}(y) = \frac{y-4}{2} \). Replace \( y \) with \( x \) to get the final inverse function: \( f^{-1}(x) = \frac{x-4}{2} \).

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Key Concepts

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

One-to-One Functions
Before understanding inverse functions, it's crucial to know what a one-to-one function is. A function is one-to-one if each output value is produced by exactly one unique input value. This means that no two different input values map to the same output value. Imagine a machine where you input a value and get an output. For it to be considered one-to-one, every input must produce a unique and different output.

One easy method to determine if a function is one-to-one is by using the horizontal line test.
Horizontal Line Test
The horizontal line test is a simple visual method to determine if a function is one-to-one. To perform this test, follow these steps:

  • Draw the graph of the function.
  • Visualize or draw horizontal lines across the graph at different output values.
  • Check if any horizontal line intersects the graph at more than one point.

If any horizontal line intersects the graph more than once, the function is not one-to-one. If every horizontal line intersects the graph at most once, the function is one-to-one.

For example, with the function \( f(x) = 2x + 4 \), it is a straight line with a non-zero slope. Each horizontal line will intersect this graph exactly once. Therefore, it passes the horizontal line test and is one-to-one.
Solving Equations to Find the Inverse
After confirming a function is one-to-one, you can find its inverse. The inverse function essentially reverses the roles of inputs and outputs. Here are the steps to find an inverse:

  • Start by replacing \( f(x) \) with \( y \) in the equation. For instance, \( f(x) = 2x + 4 \) becomes \( y = 2x + 4 \).
  • Solve the equation for \( x \) in terms of \( y \). To do this, isolate \( x \) on one side of the equation.

Let's solve for \( x \) in the equation \( y = 2x + 4 \):

\[ y - 4 = 2x \]

Next, divide both sides by 2:

\[ x = \frac{y - 4}{2} \]

Therefore, the inverse function is \( f^{-1}(y) = \frac{y - 4}{2} \). Finally, replace \( y \) with \( x \) to express the inverse function in terms of \( x \):

\( f^{-1}(x) = \frac{x - 4}{2} \).

By following these steps, you can easily find the inverse of a one-to-one function.

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Most popular questions from this chapter

Use the change-of-base rule (with either common or natural logarithms) to approximate each logarithm to four decimal places. $$ \log _{6} \sqrt[3]{5} $$

Use the change-of-base rule (with either common or natural logarithms) to approximate each logarithm to four decimal places. $$ \log _{\pi} e $$

Graph each logarithmic function. $$g(x)=\log _{1 / 4} x$$

Based on selected figures obtained during the years \(1970-2015,\) the total number of bachelor's degrees earned in the United States can be modeled by the function $$ D(x)=792,377 e^{0.01798 x} $$ where \(x=0\) corresponds to \(1970, x=5\) corresponds to \(1975,\) and so on. Approximate, to the nearest unit, the number of bachelor's degrees earned in 2015. (Data from U.S. National Center for Education Statistics.)

Solve each equation. Give exact solutions. $$ \log _{4}(2 x+8)=2 $$
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