Problem 77 - Algebra 2

Question

Find the inverse of each function. Is the inverse a function? \(f(x)=4 x-1\)

Step-by-Step Solution

Answer

The inverse function of \(f(x) = 4x - 1\) is \(f^{-1}(x) = (x+1)/4\). Yes, this inverse is a function.

Step 1: Swap x and y

The first step in finding the inverse of a function is swapping x and y. So our function y=4x-1 becomes x=4y-1.

Step 2: Solve for y

Now, we have to solve this new equation for y. This will yield y=(x+1)/4.

Step 3: Check if the inverse is a function

To check if the inverse is a function, we apply the vertical line test. Sketch the graph of the line y=(x+1)/4. A line can only pass through any vertical line at one point proving the inverse is a function.

Key Concepts

Function Notation

Understanding function notation is crucial when dealing with problems involving inverse functions. In essence, function notation uses symbols to convey a function's output in relation to its input. The most common form is denoted by letters such as \( f(x) \), where \( f \) represents the function, and \( x \) symbolizes the input. This notation tells us clearly what the input is and the corresponding output when that input is substituted.

• For example, in the function \( f(x) = 4x - 1 \), if \( x \) equals 2, then \( f(2) = 4(2) - 1 = 7 \).
• The notation \( f(x) \) helps in substituting inputs easily to find outputs, making it easier to identify patterns or transformations, especially when trying to find the inverse.

Practice reading functions with this notation. It will serve as a guide to understanding, manipulating, and finding inverses of functions.

Vertical Line Test

The vertical line test is a simple yet powerful tool to determine if a curve is the graph of a function. A graph represents a function if and only if no vertical line intersects the graph at more than one point.
This test works because functions must have only one output, or \( y \), for each input, or \( x \).

• When finding the inverse of a function, like turning \( y = 4x - 1 \) into \( x = 4y - 1 \), we solve to get \( y = \frac{x+1}{4} \).
• To verify that \( y = \frac{x+1}{4} \) is indeed a function, we graph it and apply the vertical line test.
• If a vertical line touches the graph only once at any position, the graph represents a function.

This way, we confirm that the inverse of a function is still a function, ensuring the mathematical operations preservations.

Graphing Functions

Graphing functions is instrumental in visualizing the behavior of functions and their inverses. When given the function \( f(x) = 4x - 1 \), graphing helps us understand its linearity and direction.
Here's how to effectively graph these types of functions:

• First, identify the slope and y-intercept in the function \( f(x) = mx + b \). For \( f(x) = 4x - 1 \), the slope \( m \) is 4, and the y-intercept \( b \) is -1.
• Plot the y-intercept \((-1)\) on the graph and use the slope to determine another point. A slope of 4 means you rise 4 units in y for every 1 unit you move right on the x-axis.
• Draw a straight line through these points to represent \( f(x) \). To graph the inverse, plot \( y = \frac{x+1}{4} \) in a similar fashion by identifying its slope and intercept.

Using graphs, you can visually confirm whether both original and inverse functions pass the vertical line test, assuring they are valid functions.