Function Notation

Once you know what a function is, the next step is learning how mathematicians write functions. Instead of repeatedly saying “the rule that takes $x$, squares it, and adds 1,” we give the function a name and use a compact notation. That notation is function notation, and it looks like $f(x) = x^2 + 1$. This section covers how to read it, use it, and evaluate it — skills you will need in every algebra topic from here on.

What $f(x)$ Means

The expression $f(x)$ is read “f of x”. It does not mean $f$ times $x$. The parentheses here are not multiplication — they indicate that $x$ is the input to the function named $f$.

$f(x) = 2x + 3$

This tells you three things:

1. The function’s name is $f$.
2. The input variable is $x$.
3. The rule is “multiply the input by 2, then add 3.”

When you see $f(5)$, it means “plug 5 in for $x$”:

$f(5) = 2(5) + 3 = 10 + 3 = 13$

Function Notation vs. $y =$

You already know equations like $y = 2x + 3$. Function notation and $y =$ notation describe the same relationship:

The advantage of function notation is that it names the function and makes the input explicit. When you work with multiple functions at once, naming matters.

Evaluating Functions at Numbers

To evaluate a function at a specific number, replace every occurrence of $x$ (or whatever the input variable is) with that number, then simplify.

Example 1: Evaluate $f(3)$ given $f(x) = x^2 - 4x + 7$

Replace $x$ with 3:

$f(3) = (3)^2 - 4(3) + 7 = 9 - 12 + 7 = 4$

Answer: $f(3) = 4$

Example 2: Evaluate $f(-2)$ given $f(x) = x^2 - 4x + 7$

Replace $x$ with $-2$:

$f(-2) = (-2)^2 - 4(-2) + 7 = 4 + 8 + 7 = 19$

Answer: $f(-2) = 19$

Example 3: Evaluate $f(0)$ given $f(x) = 3x^3 - x + 10$

$f(0) = 3(0)^3 - (0) + 10 = 0 - 0 + 10 = 10$

Answer: $f(0) = 10$

Notice that $f(0)$ always gives you the constant term of a polynomial — the value when the input is zero.

Evaluating Functions at Expressions

Function notation becomes especially powerful when the input is itself an expression — a variable, a sum, or another function.

Example 4: Find $f(a + 1)$ given $f(x) = 2x - 5$

Replace every $x$ with $(a + 1)$:

$f(a + 1) = 2(a + 1) - 5 = 2a + 2 - 5 = 2a - 3$

Answer: $f(a + 1) = 2a - 3$

Example 5: Find $f(3t)$ given $f(x) = x^2 + 2x$

Replace $x$ with $3t$:

$f(3t) = (3t)^2 + 2(3t) = 9t^2 + 6t$

Answer: $f(3t) = 9t^2 + 6t$

Example 6: Find $f(x + h)$ given $f(x) = x^2$

This is a classic setup that leads to the difference quotient in later courses:

$f(x + h) = (x + h)^2 = x^2 + 2xh + h^2$

Answer: $f(x + h) = x^2 + 2xh + h^2$

Different Function Names

There is nothing special about the letter $f$. Functions can be named with any letter or label:

• $g(x) = 5x - 1$ — a function named $g$
• $h(t) = t^2 + 3t$ — a function named $h$ with input variable $t$
• $P(n) = 2n + 100$ — a profit function named $P$ with input $n$
• $C(x) = 15x + 200$ — a cost function named $C$

Using descriptive names helps in applied problems. A function modeling cost might be called $C$, one modeling revenue might be $R$, and one modeling profit might be $P$.

Example 7: Given $g(x) = -3x + 12$ and $h(x) = x^2 - 1$, find $g(4)$ and $h(4)$

$g(4) = -3(4) + 12 = -12 + 12 = 0$

$h(4) = (4)^2 - 1 = 16 - 1 = 15$

Answer: $g(4) = 0$ and $h(4) = 15$

Example 8: Given $g(x) = -3x + 12$, find $g(a) - g(2)$

First, find $g(a)$:

$g(a) = -3a + 12$

Then, find $g(2)$:

$g(2) = -3(2) + 12 = -6 + 12 = 6$

Subtract:

$g(a) - g(2) = (-3a + 12) - 6 = -3a + 6$

Answer: $g(a) - g(2) = -3a + 6$

Reading Function Notation Aloud

Getting comfortable reading function notation out loud helps you think about it naturally:

When someone says “find f of 5,” they mean evaluate the function $f$ at $x = 5$.

Finding Input Given Output

Sometimes you know the output and need to find the input. This reverses the process — you solve an equation.

Example 9: If $f(x) = 4x - 9$, find $x$ when $f(x) = 15$

Set the function equal to 15 and solve:

$4x - 9 = 15$

$4x = 24$

$x = 6$

Check: $f(6) = 4(6) - 9 = 24 - 9 = 15$ Answer: $x = 6$

Example 10: If $g(x) = x^2 + 2$, find $x$ when $g(x) = 27$

$x^2 + 2 = 27$

$x^2 = 25$

$x = 5 \quad \text{or} \quad x = -5$

Check: $g(5) = 25 + 2 = 27$ and $g(-5) = 25 + 2 = 27$. Both check out Answer: $x = 5$ or $x = -5$

Real-World Application: Nursing — IV Drip Rate Function

In nursing, the drip rate for an IV infusion can be modeled as a function. The formula for drops per minute is:

$D(V) = \frac{V \times F}{T}$

where $V$ is the volume in mL, $F$ is the drip factor (drops per mL, a constant depending on the tubing), and $T$ is the time in minutes.

For tubing with a drip factor of 15 drops/mL delivering fluid over 60 minutes:

$D(V) = \frac{15V}{60} = \frac{V}{4}$

Evaluate for a 500 mL bag:

$D(500) = \frac{500}{4} = 125 \text{ drops per minute}$

Function notation makes it clear that the drip rate depends on the volume ordered. A different volume, like 250 mL, gives a different rate: $D(250) = 62.5$, which a nurse would round to 63 drops per minute since partial drops cannot be delivered.

Real-World Application: HVAC — Heating Cost Function

An HVAC contractor estimates monthly heating cost as a function of average outdoor temperature (in degrees Fahrenheit):

$C(t) = -2.5t + 250$

This means the colder it gets (lower $t$), the higher the cost:

• At $t = 30°$F: $C(30) = -2.5(30) + 250 = -75 + 250 = 175$ (cost is $175)
• At $t = 50°$F: $C(50) = -2.5(50) + 250 = -125 + 250 = 125$ (cost is $125)

What temperature makes the cost equal $200?

$-2.5t + 250 = 200 \implies -2.5t = -50 \implies t = 20$

When the average outdoor temperature is 20 degrees Fahrenheit, the estimated monthly heating cost is $200.

Common Mistakes to Avoid

1. Treating $f(x)$ as multiplication. $f(x)$ means “f of x,” not ”$f$ times $x$.” The parentheses indicate function input, not multiplication.
2. Forgetting to substitute everywhere. When evaluating $f(a + 1)$ for $f(x) = x^2 + 3x$, you must replace every $x$ with $(a + 1)$, not just the first one.
3. Dropping parentheses during substitution. Always wrap the substituted expression in parentheses. For $f(x) = x^2$ and input $-3$: $f(-3) = (-3)^2 = 9$, not $-3^2 = -9$.
4. Confusing $f(x) = 0$ with $f(0)$. The statement $f(x) = 0$ asks “for what $x$ is the output zero?” while $f(0)$ asks “what is the output when the input is zero?” These are different questions.
5. Thinking different function names mean different rules. The names $f$, $g$, $h$ are just labels. Two functions can have different names but the same rule, or the same name used in different contexts.

Practice Problems

Test your understanding with these problems. Click to reveal each answer.

Key Takeaways

• $f(x)$ is read “f of x” and means the output of function $f$ when the input is $x$ — it is not multiplication
• To evaluate a function at a value, replace every instance of the input variable with that value and simplify
• You can evaluate functions at expressions like $a + 1$ or $3t$ by substituting the entire expression (with parentheses) for $x$
• Functions can have any name — $f$, $g$, $h$, $C$, $D$ — and any input variable
• Finding an input from a known output means setting $f(x)$ equal to the output and solving the equation
• Function notation is the standard language for applied formulas in nursing, HVAC, engineering, and every other quantitative field

Return to Algebra for more topics in this section.