Functions — Core Skills

Functions: The Basics

Part 1 of 2 — The Basics

A function is a rule. You put a number in, the rule does something to it, and one number comes out.

Think of a vending machine. Press button $4$ and you get one specific snack. Press button $4$ again and you get that same snack. A function works the same way: the same number in always gives the same number out.

Reading the notation

The name $f(x)$ is read "$f$ of $x$."

• $f$ is the name of the function.
• $x$ is the number you put in.
• $f(x)$ is the number that comes out.

Important: $f(x)$ does not mean $f$ times $x$. The parentheses are holding the input, not multiplying.

Evaluating a function

To evaluate a function means to find what comes out. There is one move: wherever you see $x$, write the input number instead. Then do the arithmetic.

Worked example. If $f(x) = 3x + 2$, what is $f(4)$?

Step 1. The input is $4$, so replace $x$ with $4$:

$f(4) = 3(4) + 2$

Step 2. Multiply first: $3 \times 4 = 12$.

$f(4) = 12 + 2$

Step 3. Add: $12 + 2 = 14$.

So $f(4) = 14$.

Always multiply before you add. That order keeps you from getting $3 + 2 = 5$ and then multiplying by $4$, which would give the wrong answer.

Functions: Practice

Part 2 of 2 — Practice

The steps for evaluating

1. Find the number inside the parentheses. That is your input.
2. Rewrite the rule, putting that number everywhere $x$ appears.
3. Multiply before you add or subtract.
4. The number you end with is the output.

What the two numbers mean

Many SAT functions look like $f(x) = mx + b$. Each part has a name and a meaning, and the test asks about both.

• $m$ is the slope, also called the rate of change. It is the number multiplied by $x$, and it tells you how much the output changes each time the input goes up by one.
• $b$ is the $y$-intercept, also called the starting value. It is the number sitting by itself, and it is what you have before anything changes.

A number multiplied by a variable is also called a coefficient, so $m$ is the coefficient of $x$.

Worked example. A gym charges a \50$sign-up fee plus$$20$each month. The cost after$m$months is$C(m) = 20m + 50$.

• The $20$ is the slope. It is the monthly rate, because the cost grows by \20$ every month.
• The $50$ is the $y$-intercept. It is the starting value, because you pay it once at zero months.

When a question asks what a number "represents," check whether it is attached to the variable. Attached means rate. Standing alone means starting value.