Problem 77 - Algebra 2

Question

Find the inverse of each function. Is the inverse a function? $$ y=5 x+7 $$

Step-by-Step Solution

Answer

The inverse of the function \(y = 5x + 7\) is \(y = \frac{x - 7}{5}\). Yes, this inverse expression is also a function.

Step 1: Identifying the given function

Identify the function for which the inverse is to be calculated. Here, the function is \(y = 5x + 7\).

Step 2: Swap Variables x and y

Swap the x and y variables in the equation, which gives us \(x = 5y + 7\).

Step 3: Solve for y

To isolate y, start by subtracting 7 from both sides of the equation. This yields \(x - 7 = 5y\). Finally, divide both sides by 5. This gives us the inverse function \(y = \frac{x - 7}{5}\).

Step 4: Confirm if the inverse is a function

As a linear function, the original function \(y = 5x + 7\) passes the horizontal line test, meaning that there is only one y value for each x value in the function. Hence the inverse function \(y = \frac{x - 7}{5}\) is also a function.

Key Concepts

Linear Functions

Linear functions are one of the simplest types of functions that you'll encounter. They are characterized by a constant rate of change, which is represented by the slope of the line. The general form of a linear function is given by:

• \( y = mx + b \)

Here, \( m \) represents the slope of the line, telling us how steep the line is, while \( b \) is the y-intercept, indicating where the line crosses the y-axis.
In our exercise, the function \( y = 5x + 7 \) is linear, with a slope of 5 and a y-intercept of 7. This means for every unit increase in \( x \), \( y \) increases by 5.
Linear functions are simple yet powerful because their inverse is also a linear function, as seen in this exercise. Understanding how to manipulate these forms helps in various fields such as economics, physics, and everyday life calculations.

Horizontal Line Test

The horizontal line test is a simple method to check whether a function has an inverse that is also a function. To apply this test, you imagine drawing horizontal lines across the graph of the function. If any horizontal line intersects the graph more than once, the function does not have an inverse that is also a function.
Since our function \( y = 5x + 7 \) passes the horizontal line test, each horizontal line will intersect the graph only once. This indicates that each y-value is unique to a corresponding x-value. Thus, its inverse exists and is a function as well. The horizontal line test is crucial for determining whether a function is one-to-one, a key trait for a function to have an inverse.

Function Composition

Function composition involves combining two functions to see how one affects the other. For instance, given functions \( f(x) \) and \( g(x) \), you can create a composite function \( (f \circ g)(x) = f(g(x)) \).
One exciting application of function composition is verifying inverse functions. For two functions to be inverses, the composition of one with the other should yield the identity function \( x \). In mathematical terms:

• \( f(g(x)) = x \)
• \( g(f(x)) = x \)

For the given function \( y = 5x + 7 \) and its inverse \( y = \frac{x-7}{5} \), checking the compositions can confirm that they are truly inverses of each other. Function composition serves not only to validate inverses but also to help visualize how different processes or operations interconnect.

One-to-One Functions

A one-to-one function is an essential concept in mathematics for identifying functions with inverses. If a function is one-to-one, every x-value has a unique y-value and vice versa; no y-value in the range is mapped by more than one x-value from the domain.
The uniqueness of one-to-one functions means that their inverses will also be functions. To determine if a function is one-to-one, you can use the horizontal line test, as explained earlier. The function \( y = 5x + 7 \) qualifies as one-to-one because each y-value corresponds to exactly one x-value.
One-to-one functions are significant, particularly in fields such as computer science and cryptography, where ensuring a unique and predictable output based on input is essential. Understanding these concepts not only enhances problem-solving skills but also broadens the appreciation for how functions operate in various disciplines.