Math Help [Request Topics]

Functions and Graphing

Example 1: We are given the function f(x) = 4x^2. Solve for x when f(x) = 7.

So, we want to find what values of x lead to f(x) = 7. Our first step is to plug the value of f(x) we are given into the equation.

7 = 4x^2

Now, we need to isolate the variable we're solving for on one of the sides. In this case, I will isolate x to the right side. We do this by dividing any constants from one side over to the other side.

7/4 = (4/4) * x^2

7/4 = x^2

Now, we need to solve for x. Since we are working with x^2 we need to drop the exponent of x down to 1. We can do this by taking the square root. I'll point out the actual symbol later when I do a handwritten solution, but for now, the square root symbol will be "sqrt(...)."

sqrt(7/4) = sqrt(x^2)

It's always important to remember, whatever you do on one side of the equation, you have to do it to the other side as well. The square root of x^2 is just x, but there's a slight catch. Can you see it? If not, that's perfectly fine, I'll tell you right now. There are two possible values of x that lead to x^2 = 7/4. The first being sqrt(7/4), but the second one is -sqrt(7/4). The reason -sqrt(7/4) is an answer is because if you multiply -sqrt(7/4) to itself, you get two negative numbers multiplying together, which gives you a positive number. This means we have two answers for what x could be. We write this solution as follows.

x = sqrt(7/4), -sqrt(7/4)

The next thing we want to look at is the inverse of functions. If we are given a function f(x) we want to find the inverse of the function. That is, we want a function so that we can map values of f(x) back to the values of x. There's a very simple method to do this, and it takes very few steps.

1. Take f(x) and re-label it as y.

2. Swap the x and the y variables.

3. Solve for y.

4. (optional) replace y with f^-1(x).

To do this, we need to know the inverse of some common functions and their inverses.

You'll eventually learn about the bottom 5 functions later on. For now, all you need to worry about are the top 3 on the list. The weird-looking division sign with the small n is called the "root" symbol. The square root is written as just the symbol itself with no number written down. When you get to exponents greater than 2, you have to write the number down on the left of the root symbol. Also, we will only worry about cases right now when n > 0. 

Example: Find the inverse function of f(x) = x^2 + 5.

Solution: