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Chapter 1: Problem 68

In Exercises 63-76, determine whether the function has an inverse function. If it does, find the inverse function. \(g(x) = \frac{3x+4}{5}\)

Short Answer

Expert verified
Yes, the function \(g(x) = \frac{3x+4}{5}\) does have an inverse. The inverse function is \(g^{-1}(x) = \frac{5x - 4}{3}\).

Step by step solution

01

Check if the function is one-to-one

The function \(g(x) = \frac{3x+4}{5}\) is a linear function, since its graph is a straight line. All linear functions are one-to-one, so \(g(x)\) does have an inverse.
02

Find the inverse function

To find the inverse of a function, we replace \(g(x)\) with \(y\), so the equation now is \(y = \frac{3x+4}{5}\). Then, we switch the roles of \(x\) and \(y\), making it \(x = \frac{3y+4}{5}\). Afterwards, we solve the equation for \(y\), which would be the inverse function. By multiplying each side by 5, the equation is \(5x = 3y + 4\). Subtracting 4 from each side gives \(5x - 4 = 3y\), and finally, dividing each side by 3 gives the inverse function \(g^{-1}(x) = \frac{5x - 4}{3}\).

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Key Concepts

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

One-to-One Function
Understanding one-to-one functions is crucial when exploring inverse functions. A function is called one-to-one if different inputs (or 'x' values) always produce different outputs (or 'y' values). This ensures that each 'y' value is paired with exactly one 'x' value. A simple way to check if a function is one-to-one is by using the horizontal line test.
  • If you can draw a horizontal line through the graph of the function and it intersects the graph at most once, the function is one-to-one.
For example, the function \(g(x) = \frac{3x+4}{5}\) is a linear function. Since all linear functions have graphs that are straight lines, they naturally pass the horizontal line test. This means they are one-to-one and thus can have an inverse.
Linear Function
Linear functions are a type of function where the graph is a straight line. These functions come in the form \(f(x) = ax + b\), where \(a\) and \(b\) are constants. In the function \(g(x) = \frac{3x+4}{5}\), the arrangement is similar, and you can rearrange it to the form \(f(x) = \frac{3}{5}x + \frac{4}{5}\).
Characteristics of linear functions make them simpler to analyze, especially when dealing with inverses:
  • They have a constant rate of change, depicted by the slope \(a\). In our example, the slope is \(\frac{3}{5}\).
  • They don't curve or rotate, maintaining a straight path through their graph.
  • Their inverses are also linear, making calculations and graph interpretations straightforward.
These simple properties highlight the linear function's behavior and facilitate operations like finding inverses.
Finding Inverses
Now let's explore the steps to find the inverse of a function. Understanding how to find an inverse is important because it allows us to reverse the operation of the function.
Here's how you find the inverse:
  • First, replace the function notation \(g(x)\) with \(y\). This helps to clearly switch roles in the next step.
  • Next, interchange \(x\) and \(y\). This step is what reverses the function's operation.
  • Finally, solve for \(y\) to express it in terms of the new \(x\).
For our function \(g(x) = \frac{3x+4}{5}\), replacing and switching gives \(x = \frac{3y + 4}{5}\).
Solving for \(y\), we multiply both sides by 5, getting \(5x = 3y + 4\). By isolating \(y\), we obtain the inverse function: \(g^{-1}(x) = \frac{5x - 4}{3}\). Finding inverses is very useful in many real-world applications, enabling you to go "backwards" from outputs to inputs.

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Most popular questions from this chapter

DATA ANALYSIS: LIGHT INTENSITY A light probe is located \(x\) centimeters from a light source, and the intensity \(y\) (in microwatts per square centimeter) of the light is measured. The results are shown as ordered pairs \((x, y)\). \((30, 0.1881)\) \((34, 0.1543)\) \((38, 0.1172)\) \((42, 0.0998)\) \((48, 0.0775)\) \((50, 0.0645)\) A model for the data is \(y = 262.76/x^{2.12}\) (a) Use a graphing utility to plot the data points and the model in the same viewing window. (b) Use the model to approximate the light intensity 25 centimeters from the light source.

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