Problem 74 - Precalculus Student Solutions Manual 5th

Question

For each function as defined that is one-to-one, (a) write an equation for the inverse function in the form y = ƒ -11x2, (b) graph ƒ and ƒ -1 on the same axes, and (c) give the domain and the range of ƒ and ƒ -1. If the function is not one-to-one, say so. $$f(x)=\frac{-3 x+12}{x-6}, \quad x \neq 6$$

Step-by-Step Solution

Answer

Inverse function: \( y= \frac{6x+12}{x+3} \). Domain and range of f: all reals except 6. Domain and range of f^-1: all reals except -3.

Step 1: Check if the function is one-to-one

A function is one-to-one if every y-value corresponds to exactly one x-value. Check if the given function passes the Horizontal Line Test.

Step 2: Find the inverse function

To find the inverse function, switch the roles of x and y and solve for y. Start with the equation: \[ y = \frac{-3x + 12}{x - 6} \] and switch to \[ x = \frac{-3y + 12}{y - 6} \].

Step 3: Solve for y

Cross-multiply to get rid of the fraction: \[ x(y-6)= -3y + 12 \]. Simplify and solve for y: \[ xy - 6x = -3y + 12 \]. Rearrange to isolate y: \[ xy + 3y = 6x + 12 \]. Factor out y: \[ y(x+3) = 6x + 12 \]. Finally, solve for y: \[ y = \frac{6x + 12}{x + 3} \].

Step 4: Graph the function and its inverse

Graph the original function and its inverse on the same set of axes. The graph of the inverse function will be a reflection of the original function across the line y = x.

Step 5: Determine the domain and range of f and f^-1

The domain of the original function f(x) is all real numbers except x = 6. The range of f(x) is all real numbers except y = 3. For the inverse function f^-1(x), the domain is all real numbers except x = -3, and the range is all real numbers except y = 6.

Key Concepts

One-to-one Functions

A function is called one-to-one if each output (or y-value) is produced by exactly one input (or x-value). In other words, no two distinct inputs map to the same output. This uniqueness is crucial for a function to have an inverse. If a function is not one-to-one, its inverse will not be a function.

To determine if a function is one-to-one, you can use the **Horizontal Line Test**. If any horizontal line intersects the graph of the function at more than one point, the function is not one-to-one. For example, if we consider the function given in the exercise, we need to check whether it passes the horizontal line test to conclude if it is one-to-one.

Horizontal Line Test

The **Horizontal Line Test** is a visual way to determine if a function is one-to-one. Here’s how it works:

• Draw horizontal lines across the graph of your function.
• If any horizontal line crosses the graph more than once, the function is not one-to-one.
• If every horizontal line crosses the graph at most once, the function is one-to-one.

Let's apply this test to the function given in the exercise:
\( f(x) = \frac{-3x + 12}{x - 6} \).
If no horizontal line intersects the graph of this function more than once, then we can confirm that it is one-to-one.

Domain and Range

The **domain** of a function is the set of all possible input values (x-values) for which the function is defined. The **range** is the set of all possible output values (y-values) that the function can produce.
In the given exercise, the domain and range of both the original function and its inverse are critical to understand:

• For the original function \( f(x) \), we have:
  • Domain: All real numbers except \( x = 6 \)
  • Range: All real numbers except \( y = 3 \)
• For the inverse function \( f^{-1}(x) \), we get:
  • Domain: All real numbers except \( x = -3 \)
  • Range: All real numbers except \( y = 6 \)

Understanding the domain and range helps avoid undefined mathematical operations and ensures proper graphing of both functions.