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Chapter 2: Problem 66

Find the inverse of each function and graph both \(f\) and \(f^{-1}\) on the same coordinate plane. $$f(x)=\sqrt{x-3}$$

Short Answer

Expert verified
The inverse function is \( f^{-1}(x) = x^2 + 3 \). Graph both functions for a visual representation.

Step by step solution

01

- Identify the function

The given function is \( f(x) = \sqrt{x - 3} \).
02

- Replace \( f(x) \) with \( y \)

Start by setting \( y = \sqrt{x - 3} \).
03

- Swap \( x \) and \( y \)

To find the inverse function, swap the variables \( x \) and \( y \). The equation becomes \( x = \sqrt{y - 3} \).
04

- Solve for \( y \)

Solve the equation \( x = \sqrt{y - 3} \) for \( y \). First, square both sides to get rid of the square root: \( x^2 = y - 3 \).
05

- Isolate \( y \)

Add 3 to both sides to isolate \( y \): \( y = x^2 + 3 \).
06

- Write the inverse function

Thus, the inverse function is \( f^{-1}(x) = x^2 + 3 \).
07

- Graph both functions

Graph the original function \( f(x) = \sqrt{x - 3} \) and the inverse function \( f^{-1}(x) = x^2 + 3 \) on the same coordinate plane. Remember that the graph of the inverse function is a reflection of the graph of the original function across the line \( y = x \).

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Key Concepts

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

Finding the inverse of a function
When finding the inverse of a function, you're essentially reversing the process of the original function. The steps to find the inverse function ensure that for every input-output pair in the original function, the inverse will swap these roles.
Here are the core steps in depth:
  • Start by identifying the given function. In this example, it's given as \(f(x) = \sqrt{x - 3}\).
  • Replace \(f(x)\) with \(y\). So, \(y = \sqrt{x - 3}\).
  • Swap the roles of \(x\) and \(y\), resulting in the equation \(x = \sqrt{y - 3}\).
  • Solve this swapped equation for \(y\). To do this, square both sides to eliminate the square root: \(x^2 = y - 3\). Then, isolate \(y\) by adding 3 to both sides: \(y = x^2 + 3\).
  • The equation you now have represents the inverse function. For this example, the inverse is \(f^{-1}(x) = x^2 + 3\).
Graphing functions
Graphing functions involves plotting them on a coordinate plane to visualize their behavior. For this exercise, you need to graph both the original function and its inverse.
Let's break this down:
  • Plot the original function \(f(x) = \sqrt{x - 3}\). This represents a square root function that's been shifted to the right by 3 units.
  • Next, plot the inverse function \(f^{-1}(x) = x^2 + 3\). This is a parabolic function that opens upwards and is shifted upwards by 3 units.
  • Use different colors or styles for each graph to easily distinguish between the original function and its inverse.
The visualization will help you see how each input of the original function maps to an output, and how the inverse function maps back in reverse.
Reflection over the line y=x
When graphing an original function and its inverse, another important concept to understand is the reflection over the line \(y = x\). This line is often used as a reference because the graph of a function and its inverse are reflections across it.
Consider these key points:
  • The line \(y = x\) bisects the first and third quadrants of the coordinate plane at a 45-degree angle.
  • If you were to fold the graph along this line, the original function's graph would match the inverse function's graph.
  • In our example, you'll notice that for every point \((a, b)\) on the graph of the original function, there's a corresponding point \((b, a)\) on the inverse graph.
Understanding this reflection helps verify that you've correctly found and graphed the inverse function. It's a visual confirmation that the two functions are indeed inverses of each other.

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