Functions and Function Notation

What Is a Function?

A function is a rule that assigns exactly one output to each input. Think of it as a machine: you feed in a number (the input), the machine processes it according to its rule, and it produces exactly one number (the output).

If $f(x) = 2x + 3$, then for every value of $x$ there is exactly one value of $f(x)$:

• Input $x = 1$ → Output $f(1) = 2(1) + 3 = 5$
• Input $x = -4$ → Output $f(-4) = 2(-4) + 3 = -5$

The notation $f(x)$ is read "f of x." The $x$ inside the parentheses is the input (also called the argument or independent variable), and $f(x)$ is the output (also called the dependent variable or the value of the function at $x$). The letter $f$ is simply the name of the function — other common names are $g$, $h$, $p$, etc.

Critical point: $f(x)$ does NOT mean "$f$ times $x$." It is not multiplication. It means "the function named $f$ evaluated at the input $x$." This notational confusion trips up many students.

Evaluating Functions

To evaluate $f(a)$, replace every $x$ in the function's formula with $a$, then simplify.

If $f(x) = x^2 - 3x + 5$:

• $f(2) = (2)^2 - 3(2) + 5 = 4 - 6 + 5 = 3$
• $f(-1) = (-1)^2 - 3(-1) + 5 = 1 + 3 + 5 = 9$
• $f(0) = (0)^2 - 3(0) + 5 = 0 - 0 + 5 = 5$
• $f(a + 1) = (a+1)^2 - 3(a+1) + 5 = a^2 + 2a + 1 - 3a - 3 + 5 = a^2 - a + 3$

The last example shows that you can evaluate a function at an algebraic expression, not just a number. The ACT tests this — for instance, asking you to find $f(2x)$ or $f(x+h)$.

Evaluating Functions from Tables and Graphs

The ACT frequently presents functions as tables or graphs rather than equations.

From a table: If a table shows $f(3) = 7$, then the input 3 produces the output 7. To find $f(3)$, look up $x = 3$ and read the corresponding $f(x)$ value. To find $x$ when $f(x) = 7$, look for 7 in the output column and read the corresponding $x$.

From a graph: $f(a)$ is the $y$-coordinate of the graph at $x = a$. To find $f(3)$, go to $x = 3$ on the horizontal axis, move up (or down) to the curve, and read the $y$-value.

Domain and Range

The domain is the set of all valid inputs ($x$-values for which the function is defined). The range is the set of all possible outputs ($y$-values or $f(x)$-values that the function actually produces).

Common domain restrictions:

• Division by zero: The denominator cannot equal zero. For $f(x) = \frac{1}{x-3}$, the domain excludes $x = 3$.
• Square roots of negatives: The expression under a square root must be $\geq 0$. For $f(x) = \sqrt{x - 2}$, the domain is $x \geq 2$.
• Logarithms: The argument must be positive. For $f(x) = \ln(x + 4)$, the domain is $x > -4$.
• No restrictions: Polynomials like $f(x) = x^2 + 3x - 1$ have domain = all real numbers.

Finding the range: Consider what output values are possible.

• For $f(x) = x^2$: the range is $y \geq 0$ (squaring always gives non-negative results)
• For $f(x) = -x^2 + 4$: the maximum value is 4 (at $x = 0$), so range is $y \leq 4$
• For $f(x) = |x|$: the range is $y \geq 0$

Composition of Functions

Composition means applying one function and then feeding the result into another function. The notation is $f(g(x))$ or $(f \circ g)(x)$, which means: first apply $g$ to $x$, then apply $f$ to the result.

If $f(x) = 2x + 1$ and $g(x) = x^2$:

Numerical composition:

• $f(g(3)) = f(3^2) = f(9) = 2(9) + 1 = 19$
• $g(f(3)) = g(2(3)+1) = g(7) = 7^2 = 49$

Notice: $f(g(3)) \neq g(f(3))$. The order of composition matters!

Algebraic composition: $f(g(x)) = f(x^2) = 2(x^2) + 1 = 2x^2 + 1$ $g(f(x)) = g(2x+1) = (2x+1)^2 = 4x^2 + 4x + 1$

Again, different results depending on the order.

Function Transformations

Understanding how changes to the equation affect the graph is a crucial and frequently tested ACT skill:

Transformation	Effect on Graph
$f(x) + k$	Shift up $k$ units
$f(x) - k$	Shift down $k$ units
$f(x + h)$	Shift left $h$ units
$f(x - h)$	Shift right $h$ units
$-f(x)$	Reflect over the $x$-axis (flip vertically)
$f(-x)$	Reflect over the $y$-axis (flip horizontally)
$af(x)$ where $a > 1$	Vertical stretch by factor $a$
$af(x)$ where $0 < a < 1$	Vertical compression by factor $a$

The counterintuitive rule: $f(x + h)$ shifts left, not right. The sign inside the parentheses is opposite to the direction of the horizontal shift. This is because adding $h$ to $x$ makes the function reach the "same value" sooner (earlier on the $x$-axis).

How to remember: Vertical changes ($+k$, $-k$, multiplication by $a$) do what you expect. Horizontal changes ($+h$, $-h$ inside the argument) do the opposite of what you expect.

Inverse Functions

The inverse function $f^{-1}(x)$ "undoes" what $f$ does. If $f(a) = b$, then $f^{-1}(b) = a$. The function and its inverse "cancel each other out."

To find the inverse algebraically:

1. Replace $f(x)$ with $y$
2. Swap $x$ and $y$
3. Solve for $y$
4. Rename $y$ as $f^{-1}(x)$

Example: Find the inverse of $f(x) = 3x - 7$. $y = 3x - 7$ $x = 3y - 7 \quad \text{(swap } x \text{ and } y\text{)}$ $x + 7 = 3y$ $y = \frac{x + 7}{3}$ $f^{-1}(x) = \frac{x + 7}{3}$

Verify: $f(f^{-1}(x)) = 3 \cdot \frac{x+7}{3} - 7 = (x+7) - 7 = x$ ✓

Quick method for finding $f^{-1}(c)$: Instead of finding the full inverse formula, set $f(x) = c$ and solve for $x$. For example, to find $f^{-1}(12)$ when $f(x) = 5x - 3$: solve $5x - 3 = 12$, getting $x = 3$. So $f^{-1}(12) = 3$.

Even and Odd Functions

• Even functions: $f(-x) = f(x)$ for all $x$. The graph is symmetric about the $y$-axis. Examples: $f(x) = x^2$, $f(x) = |x|$, $f(x) = \cos x$.
• Odd functions: $f(-x) = -f(x)$ for all $x$. The graph is symmetric about the origin. Examples: $f(x) = x^3$, $f(x) = x$, $f(x) = \sin x$.
• Neither: Most functions are neither even nor odd. Example: $f(x) = x^2 + x$.

Piecewise Functions

A piecewise function uses different rules for different intervals of $x$:

$f(x) = \begin{cases} 2x + 1 & \text{if } x < 0 \\ x^2 & \text{if } x \geq 0 \end{cases}$

To evaluate $f(-3)$: since $-3 < 0$, use the first rule: $f(-3) = 2(-3) + 1 = -5$.

To evaluate $f(4)$: since $4 \geq 0$, use the second rule: $f(4) = 4^2 = 16$.

Always check which interval your input falls in before applying a rule.

Tip 1: Substitute Carefully with Parentheses

The most common function questions simply ask you to evaluate $f(a)$. The key is using parentheses when substituting, especially with negative numbers. Write $f(-3) = (-3)^2 = 9$, not $f(-3) = -3^2 = -9$. Those parentheses make all the difference.

Tip 2: For Composition, Work Inside-Out

In $f(g(2))$, first compute $g(2)$, then plug the result into $f$. Always start with the innermost function. Do not try to combine steps — that is where errors creep in.

Tip 3: Remember Transformation Directions

Horizontal shifts are counterintuitive: $f(x - 3)$ shifts right 3, and $f(x + 3)$ shifts left 3. Vertical shifts are intuitive: $f(x) + 3$ shifts up 3. If you ever get confused, test with a specific point: if $(2, 5)$ is on $f(x)$, then $(5, 5)$ is on $f(x-3)$ (shifted right 3).

Tip 4: Use the Vertical Line Test for Graphs

If the ACT shows a graph and asks "which could be $f(x)$?", apply the vertical line test. If any vertical line crosses the graph more than once, it is not a function (each input must have exactly one output).

Tip 5: Plug In for Abstract Function Questions

When a question involves abstract expressions like $f(x+h) - f(x)$ or asks about properties of functions, assign $f$ a simple specific function like $f(x) = x^2$ and compute concrete numbers to match answer choices.