Problem 26 - Algebra A Combined Function

Question

Each of the following functions is one-to-one. Find the inverse of each function and graph the function and its inverse on the same set of axes. $$ f(x)=4 x+9 $$

Step-by-Step Solution

Answer

The inverse function is \( f^{-1}(x) = \frac{x - 9}{4} \).

Step 1: Replace f(x) with y

Start by replacing the function notation \( f(x) \) with \( y \) to simplify the equation.\[ y = 4x + 9 \]

Step 2: Swap x and y

Next, to find the inverse of the function, swap \( x \) and \( y \). This is the defining characteristic of inverse functions.\[ x = 4y + 9 \]

Step 3: Solve for y

Isolate \( y \) by solving the equation from the previous step:1. Subtract 9 from both sides to get: \[ x - 9 = 4y \]2. Divide both sides by 4: \[ y = \frac{x - 9}{4} \]

Step 4: Write the inverse function

Now express the inverse function using function notation. Therefore, the inverse of \( f(x) = 4x + 9 \) is denoted as \( f^{-1}(x) \).\[ f^{-1}(x) = \frac{x - 9}{4} \]

Step 5: Graph the function and its inverse

To graph \( f(x) = 4x + 9 \) and its inverse \( f^{-1}(x) = \frac{x - 9}{4} \), plot both equations on the same set of axes. They should be symmetrical with respect to the line \( y=x \). The line \( y=x \) acts as a mirror line for the function and its inverse.

Key Concepts

One-to-One Functions

A one-to-one function is a type of function where each output value is linked to only one input value, and vice versa. This means the function has a unique property: no two different inputs will map to the same output. This is essential for a function to have an inverse.

• Each y-value corresponds to exactly one x-value.
• One-to-one functions pass the Horizontal Line Test, where any horizontal line intersects the graph at most once.

Identifying a function as one-to-one is a crucial first step before attempting to find its inverse.

Function Notation

Function notation is a shorthand way to express equations compactly and precisely. By using letters and symbols, we easily denote the relationships between input and output values.

• For example, \( f(x) = 4x + 9 \) indicates a relationship where when you input \( x \), you get \( 4x+9 \).
• Generalized by \( f(x) \), with \( f \) representing the function and \( x \) the variable.

Function notation simplifies expressing inverse functions. To find inverses, we use different notation, like \( f^{-1}(x) \), to represent the inverse relationship.

Graphing Inverse Functions

Graphing inverse functions involves plotting both the original function and its inverse on the same axes. This provides visual insight into their relationship.

• For the original function \( f(x) = 4x + 9 \), the graph is a straight line with a slope of 4 and a y-intercept of 9.
• The inverse, \( f^{-1}(x) = \frac{x - 9}{4} \), also plots as a line with a different slope and intercept.

To graph them:- Take several points from the original function.- Swap the x and y coordinates for the inverse.- Plot them to visualize both lines on the same axes.

Symmetry in Graphs

Symmetry in graphs of functions and their inverses reveals a mirrored relationship. Specifically, when graphing a function and its inverse, you notice they are mirrored over the line \( y = x \).

• This line serves as the axis of symmetry.
• When a point \((a, b)\) is on the graph of the function \( f(x) \), \((b, a)\) will appear on the graph of its inverse.

Symmetry helps verify the correctness of plotted inverses and reinforces the visual intuition that both functions reciprocally return each other's outputs.