# Which Functions Have Inverses that are Functions?

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Have you ever wondered which function has an inverse that is also a function? It’s an important question to ask when dealing with mathematical functions, as knowing whether or not a function has an inverse that is also a function can greatly impact the calculations and analyses you perform.

In general, a function has an inverse that is also a function if and only if it passes the horizontal line test. This means that if any horizontal line intersects the graph of the function at more than one point, then the function does not have an inverse that is also a function. So, which of the functions b(x) = x2 + 3, d(x) = –9, m(x) = –7x, and p(x) = |x| pass the horizontal line test and therefore have an inverse that is also a function?

## What is an Inverse Function?

An inverse function is a function that “undoes” another function. In other words, if we apply a function to a value and then apply its inverse function to the result, we should get back the original value. For example, if we apply the function f(x) = 2x to the value 3, we get 6. If we then apply its inverse function, denoted by f^-1(x), to 6, we should get back 3.

Not all functions have an inverse function. For a function to have an inverse function, it must satisfy the horizontal line test. This means that every horizontal line intersects the graph of the function at most once. If a horizontal line intersects the graph of the function more than once, then the function does not have an inverse function.

Functions that have an inverse function are called one-to-one functions. One-to-one functions are also called injective functions, because they “inject” each input value to a unique output value.

In the case of the functions b(x) = x^2 + 3, d(x) = -9, m(x) = -7x, and p(x) = |x|, only one of them has an inverse function that is also a function. Can you guess which one? Let’s take a closer look at each of them:

• The function b(x) = x^2 + 3 is not one-to-one, because it fails the horizontal line test. For example, the horizontal lines y = 4 and y = 2 both intersect the graph of b(x) at two different points. Therefore, b(x) does not have an inverse function that is a function.

• The function d(x) = -9 is a constant function, which means that it maps every input value to the same output value. Constant functions do not have an inverse function that is a function, because they fail the horizontal line test. In the case of d(x) = -9, every horizontal line intersects the graph of d(x) at the same point (-9).

• The function m(x) = -7x is a linear function with a negative slope. Linear functions with nonzero slopes are one-to-one, so m(x) has an inverse function. The inverse function of m(x) is given by m^-1(x) = -x/7.

• The function p(x) = |x| is not one-to-one, because it fails the horizontal line test. For example, the horizontal lines y = 1 and y = 0 both intersect the graph of p(x) at two different points. Therefore, p(x) does not have an inverse function that is a function.

In summary, only the function m(x) = -7x has an inverse function that is also a function.

## Which Functions Have Inverses?

When we talk about inverse functions, we are referring to functions that undo the operation of another function. In other words, if we apply a function to a value and then apply its inverse function to the result, we should get back the original value. However, not all functions have inverses that are also functions. In this section, we will examine four different functions and determine which ones have inverses that are also functions.

### b(x) = x2 + 3

The function b(x) = x2 + 3 is a quadratic function that opens upwards and has a vertex at (0,3). This function is one-to-one and has an inverse function that is also a function. To find the inverse function, we can switch the roles of x and y and solve for y.

x = y2 + 3

y2 = x – 3

y = ±√(x – 3)

Since we want the inverse function to also be a function, we need to restrict the domain of b(x) to only include non-negative values of x. Therefore, the inverse function of b(x) is:

b⁻¹(x) = √(x – 3), x ≥ 3

### d(x) = –9

The function d(x) = –9 is a constant function that always returns the value –9. This function is not one-to-one and does not have an inverse function that is also a function. This is because multiple values of x will map to the same output value of –9, making it impossible to determine a unique inverse for each input value.

### m(x) = –7

The function m(x) = –7 is also a constant function that always returns the value –7. Like d(x), this function is not one-to-one and does not have an inverse function that is also a function. No matter what value of x we choose, the output will always be –7, making it impossible to determine a unique inverse for each input value.

### p(x) = |x|

The function p(x) = |x| is an absolute value function that reflects the input value across the y-axis and then takes the absolute value of the result. This function is one-to-one and has an inverse function that is also a function. To find the inverse function, we can split the domain of p(x) into two parts: x ≥ 0 and x < 0.

For x ≥ 0, the inverse function is:

p⁻¹(x) = x, x ≥ 0

For x < 0, the inverse function is:

p⁻¹(x) = –x, x < 0

Therefore, the inverse function of p(x) is:

p⁻¹(x) = x, x ≥ 0

p⁻¹(x) = –x, x < 0

## How to Find the Inverse of a Function

Finding the inverse of a function is a crucial part of mathematics, especially when dealing with functions that are not one-to-one. An inverse function is a function that “undoes” the original function, meaning that if you apply the inverse function to the output of the original function, you will get back the input value.

To find the inverse of a function, you need to follow a few simple steps:

1. Replace the function notation with “y”. For example, if you have the function f(x) = x^2, replace it with y = x^2.

2. Swap the x and y variables. This means that you now have an equation with “x” on one side and “y” on the other side. For example, if you had y = x^2, you would now have x = y^2.

3. Solve for y. This step involves isolating the y variable on one side of the equation. For example, if you had x = y^2, you would take the square root of both sides to get y = +/- sqrt(x).

4. Replace “y” with the inverse notation “f^-1(x)”. This means that you have now found the inverse function. For example, if you had y = +/- sqrt(x), you would replace “y” with “f^-1(x)” to get f^-1(x) = +/- sqrt(x).

It’s important to note that not all functions have an inverse that is also a function. A function must be one-to-one in order for it to have an inverse function. This means that each input value must correspond to a unique output value, and no two input values can have the same output value.

Out of the functions listed, only b(x) = x^2 + 3 and p(x) = |x| have an inverse that is also a function. The functions d(x) = -9 and m(x) = -7x are not one-to-one, so they do not have an inverse that is a function.

In summary, finding the inverse of a function involves swapping the x and y variables, solving for y, and replacing “y” with the inverse notation “f^-1(x)”. It’s important to note that not all functions have an inverse that is also a function, and a function must be one-to-one in order for it to have an inverse function.

## Why is it Important to Know Which Functions Have Inverses?

Understanding which functions have inverses that are also functions is important in many areas of mathematics and science. Inverse functions play a vital role in calculus, where they are used to find derivatives and integrals of functions. They are also used in geometry, physics, engineering, and many other fields.

Knowing which functions have inverses that are also functions can help us solve problems more efficiently and accurately. For example, if we need to find the derivative of a function, we can use the inverse function to do so. Similarly, if we need to find the area under a curve, we can use the inverse function to find the limits of integration.

One of the most important reasons why it is important to know which functions have inverses that are also functions is that it helps us avoid errors. If we try to find the inverse of a function that does not have an inverse, we may end up with incorrect results. This can lead to mistakes in calculations, which can have serious consequences in some situations.

For example, in engineering and physics, incorrect calculations can lead to structural failures, accidents, and other dangerous situations. In finance, incorrect calculations can lead to financial losses and other negative consequences. Therefore, it is important to be aware of which functions have inverses that are also functions, and to use them appropriately in our calculations.

In conclusion, understanding which functions have inverses that are also functions is an important concept in mathematics and science. It can help us solve problems more efficiently and accurately, avoid errors, and prevent serious consequences in some situations. By being aware of this concept, we can improve our problem-solving skills and make more informed decisions in various fields.

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