Nonlinear Equations & Functions — Core Skills

Functions: The Basics

Part 1 of 2 — One Skill, One Idea

A function is a machine. You put a number in. The machine does some arithmetic. A number comes out.

The name of the machine is usually $f$. When you see $f(x)$, read it out loud as "f of x." It does not mean $f$ times $x$. It means "the number that comes out when you put $x$ in."

The one move: replace $x$ with the number

If a question says $f(3)$, the $3$ is what you put in. Everywhere you see an $x$ in the rule, you write $3$ instead. Then you do the arithmetic.

Worked example

Say the rule is $f(x) = x^{2} + 5$. Find $f(4)$.

Step 1 — Write the rule down: $f(x) = x^{2} + 5$.

Step 2 — Replace every $x$ with $4$: $f(4) = 4^{2} + 5$.

Step 3 — Do the exponent first. An exponent tells you how many times to multiply a number by itself, so $4^{2}$ means $4 \times 4 = 16$.

Step 4 — Finish the arithmetic: $16 + 5 = 21$.

So $f(4) = 21$. That is the whole skill.

Two small things that trip people up

Negatives inside a square. $(-3)^{2}$ means $(-3) \times (-3)$. A negative times a negative is positive, so $(-3)^{2} = 9$, not $-9$.

Absolute value bars. The bars in $|x|$ mean "distance from zero," so the answer is never negative. $|-5| = 5$ and $|5| = 5$.

Functions: Practice

Part 2 of 2 — Run the Steps

Here are the steps again, in order:

1. Write down the rule, exactly as given.
2. Replace every $x$ in the rule with the number in the parentheses.
3. Do exponents first, then multiply, then add or subtract.
4. Write your answer as a single number.

Three extra facts worth memorizing

These show up again and again, and each one is a one-line fact.

Shifting a graph up or down. Adding a number on the outside moves the whole graph up. $f(x) + 3$ is the graph of $f$ moved up $3$ units. Subtracting moves it down.

Vertex form. When you see $f(x) = (x - h)^{2} + k$, the lowest point of the graph, called the vertex, sits at $(h, k)$. Watch the sign: $(x - 5)^{2} + 2$ has its vertex at $(5, 2)$, because $x - 5$ matches $x - h$ when $h = 5$.

Dividing by zero is not allowed. In a fraction like $\frac{1}{x - 6}$, the bottom cannot equal zero. Set the bottom equal to zero and solve: $x - 6 = 0$ gives $x = 6$. So $x = 6$ is the one value the function cannot use.