Finding the inverse of a function is a important math concept that comes up frequently in math, everyday life, and many disciplines.

Math majors often encounter the inverse of a function in their **undergraduate linear algebra course**. For example, in courses on linear systems, one has to find an equation whose solution is the same as the given vector. This solved equation is the inverse of the given equation representing the vector.

Every discipline has different applications for infinitesimals and infinite quantities, both of which require finding the inverse of a function. In chemistry, for example, an **igneous rock may melt** at a certain temperature, which is the inverse of its freezing point.

This article will discuss how to find the inverse of a **function using easy examples**. Before reading this article, read how to simplify functions first.

## First, isolate y by taking the square root of both sides

Next, check if the value of y is a **real number** or not by checking if it has a positive and negative value.

If the y value of the inverse equation can have only positive values, then the equation cannot be solved as it is. If the y value of the inverse equation can have only **negative values**, then the equation cannot be solved as it is.

If the y value of the inverse equation can have both positive and negative values, then solve the equation to find its solutions. These solutions are found by taking the opposite of each solution and putting them together. This solution is called “opposites” because they are solving for y when given x.

There are some cases where this *problem cannot* be solved, so check before trying to solve it.

## y = (x sqrt)(x-7)

The above equation can be simplified by factoring the denominator. By doing so, you will see that the denominator can be equated to x – 7, which makes this step valid.

By pulling out the square root of both sides, you can see that y is equal to x + 7. Now that you know this, you can solve for the inverse of y = * x2 – 7* by putting in x = -7 or x = 7.

You just learned how to find the inverse of any linear function! The hardest part is figuring out how to simplify the equation, which takes a little bit of thinking.

Did you notice that solving for y = x2 – 7 only gives you y when x is -7 or 7? This is because when solving for y, the answer would be 0 since y would be equal to **0x – 7**, and there is no **number 0x – 7**.

## Simplify and find a pattern

In this case, the **blogger suggests simplifying** the equation by finding the inverse of X = Y – 1. Then, he suggests simplifying that new equation to find a pattern.

By inverting the Y = X2 – 7 equation, you can find what X equals when you know what Y is. For example, if Y is 2, then X is *2 squared minus 7*, or 2*2-7=0.

By simplifying the new equation (X=0 when Y=2), you can see that there are two values of X that make Y equal to 2. These are -1 and 1.

This can be confusing, so let’s look at another example: If Y is 3, then X is *3 squared minus 7*, or 3*3-7=1.

## y= (x sqrt)(x-7)= (x sqrt)(x+3)

6) Finally, take the inverse using the formula Y = X^-1= {(X^2)-1}/{(X-Y)}

7) The final answer is Y = (X+3)/(X-Y)= X+6/X-1

8) This simplified equation shows that when X=1, Y=-6 and when X=-1 then Y=6

9) Inverses have many applications in mathematics, physics and engineering

The solution to this problem is that it is not possible to simplify this equation into an easier form because there isn’t a common factor between both terms. However, you will notice that if you substitute one value for another term in this equation it will result in one of two values or -6 or 6.

Solution 1: Substitute 1 for x in the original expression for y.

“y=(xxsqrt)(xxsqrt)-7=(xxsqrt)(xx)+7=(xx)+((x+3))(xs)”

In mathematics, the inverse of a function is another function that, when applied to the same input, yields the same output.

There are several ways to find the inverse of a function, and one of the simplest is to take the negation (or opposite) of each element of the function’s output, then take the sum of these new elements.

For example, let’s look at the equation y = x2 – 7. We can determine that its inverse is y = x2 + 7. To find this inverse, we would take the negation (or opposite) of each element in y = x2 – 7 and then add these new elements together.

In this case, we would negate each element in y = x2 – 7 by taking the 2 and switching it for -2, taking -7 and switching it for 7, then adding these two numbers together to get 0.

We can now say that when y = x2 + 7 is applied to any value of x, it will yield a value of y = x2 + 7.

Inverses have many applications in mathematics, physics and engineering so knowing how to find them is an important skill.