Finding Inputs of Functions

In this section, we will learn how to find the input $x$ for a function when you are given the output$f(x)$. This is like working backwards—now, instead of finding the output when you know the input, we'll figure out what input produces a specific output.

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What Does Finding the Input Mean?

When we find the input of a function, we are solving for $x$ when we know the output $f(x)$. In other words, we know the result of the function, and our job is to figure out what input value $x$ gave that result.

Imagine you've been given the answer to a puzzle, and now you need to figure out how you got there!

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Steps to Find the Input

Let's break down the steps you need to follow to find the input of a function. We'll go through an example step by step to make it easier to understand.

Step 1: Understand the Problem

First, look at the function and the output you are given. For example, let's use the function:

$f(x) = 5x - 3$

And let's say we want to find the value of $x$ that makes $f(x) = 12$. This means that when we use a certain number (the input) in this function, the result is $12$. We need to figure out what that number is.

Step 2: Set Up the Equation

To find the input, we start by letting the function equal the output we are given. But why do we do this?

Think of it this way: The function $f(x) = 5x - 3$ is like a machine that takes the input $x$, does some calculations, and gives us the output. If we know the output (in this case, $12$), we can figure out what input $x$ went into the machine by solving the equation.

So, by setting the function equal to the output, we're saying, "I know the machine gives me $12$—now, what $x$ makes that happen?"

For this example, we set up the equation:

$5x - 3 = 12$

This equation shows that when you use the function $5x - 3$, it should equal $12$. Now we need to solve for $x$ to find out what input gives us this output.

Step 3: Solve for $( x )$

Now, let's solve the equation step by step.

1. Get rid of the $-3$: To isolate $x$, we first need to get rid of the $-3$. We do this by adding $3$ to both sides of the equation: $5x - 3 + 3 = 12 + 3$ Simplifying this gives us: $5x = 15$
2. Solve for $x$: Now that we have $5x = 15$, we can find $x$ by dividing both sides by $5$: $\frac{5x}{5} = \frac{15}{5}$ Simplifying this gives us: $x = 3$ So, the input that gives $f(x) = 12$ is $x = 3$.

Step 4: Check Your Answer

It's always a good idea to double-check your work. You can do this by substituting your answer back into the original function.

For example:

• If $x = 3$, then: $f(3) = 5(3) - 3 = 15 - 3 = 12$ This confirms that our answer is correct!

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Key Points to Remember

• Set the function equal to the given output: Start by creating an equation where the function equals the output you're given. This helps you work backward to find the input.
• Solve for $x$ : Use simple steps to solve the equation and find the input value $x$.
• Check your answer: Once you've found $x$, plug it back into the function to make sure it gives you the correct output.

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