AP Precalculus: Function Concepts & Formulas
1. Domain and Range
Domain: set of all valid input values (\(x\)) for \(f(x)\)
Range: set of all possible output values (\(f(x)\))
- For \(f(x)\) rational: exclude values making denominator zero
- For \(f(x)\) with square roots: radicand must be \(\geq 0\)
- Example: \(f(x) = x^2 - 1\), Domain: \((-\infty, \infty)\), Range: \([-1, \infty)\)
- Interval notation, set-builder, & inequalities all describe domain/range
Examples:
\(f(x) = \frac{1}{x}\) → Domain: \(x \neq 0\)
\(f(x) = \sqrt{x-5}\) → Domain: \(x \geq 5\)
2. Identify Functions
A function assigns every input exactly one output.
Vertical Line Test: A graph is a function if no vertical line crosses it more than once.
Example: \(f(x) = x^2\) ✔️ is a function
\(x^2 + y^2 = 1\) (circle) ❌ is not a function (fails vertical line test)
3. Evaluate Functions
Substitute input into the function expression.
Formula: \(f(a) =\) value of \(f\) when \(x = a\)
Example: \(f(x) = 2x + 3,\ \ f(4) = 2(4)+3 = 11\)
\(f(a+b)\): replace \(x\) with \(a+b\): \(2(a+b)+3\)
4. Find Values Using Function Graphs
To find \(f(a)\) on a graph, locate \(x=a\) and read the y-value (\(f(a)\)).
Example: For a graphed function, if at \(x=2\), \(y=5\), then \(f(2) = 5\).
5. Complete a Table for a Function Graph
Fill in the missing table entries for various \(x\) using \(f(x)\).
Example: \(f(x) = 2x + 1\)
\(\begin{array}{c|c}
x & f(x) \\
\hline
1 & 3 \\
2 & 5 \\
3 & 7 \\
\end{array}\)
6. Identify Graphs: Word Problems
Interpret a graph in context of a real-world situation.
Match key graph features (intercepts, maxima/minima, intervals) to problem details.
Example: If a graph shows profit over time, maximum point = peak profit.
7. Add, Subtract, Multiply, Divide Functions
- \((f + g)(x) = f(x) + g(x)\)
- \((f - g)(x) = f(x) - g(x)\)
- \((f \cdot g)(x) = f(x) \cdot g(x)\)
- \(\left(\frac{f}{g}\right)(x) = \frac{f(x)}{g(x)},\ g(x) \neq 0\)
Example: If \(f(x) = x^2,\ g(x) = x+1\)
- \((f + g)(x) = x^2 + x + 1\)
- \((f - g)(x) = x^2 - (x + 1)\)
- \((f \cdot g)(x) = x^2(x+1)\)
- \(\left(\frac{f}{g}\right)(x) = \frac{x^2}{x+1}\) for \(x \neq -1\)
8. Composition of Functions
The output of one function becomes the input of another: \((f \circ g)(x) = f(g(x))\)
Order matters!
Example:
\(f(x) = 2x + 1\), \(g(x) = x^2 \)
\((f \circ g)(x) = f(g(x)) = 2(x^2) + 1 = 2x^2 + 1\)
\((g \circ f)(x) = g(f(x)) = (2x + 1)^2 = 4x^2 + 4x + 1\)