Finding the Solutions to a Function
Master every method for finding zeros, solving equations, and locating intersections — the skills that appear on nearly every SAT Math section.
Algebra
SAT Math
High-Yield Topic
1. What Are "Solutions" to a Function?
A solution is an $x$-value that makes an equation true. Depending on the problem, you may be asked to find:
| Problem Type | What You Solve | Graphical Meaning |
|---|---|---|
| Zeros / Roots | $f(x)=0$ | Where the graph crosses the $x$-axis |
| Specific Value | $f(x)=k$ | Where the graph meets the horizontal line $y=k$ |
| Intersection | $f(x)=g(x)$ | Where two graphs cross each other |
SAT Vocabulary: "zeros," "roots," "solutions," and "$x$-intercepts" all refer to the $x$-values where $f(x)=0$.
Why Solutions Matter
- 📍 $x$-intercepts of graphs
- 💰 Break-even points in business
- ⚖️ Equilibrium in science
- 🕐 Answering "when" and "what value" questions
- 🎯 SAT favorite: appears in 5–8 questions per test
Interactive: The Three Types of Solution Problems
Interactive visualization — available in the full lesson
2. Linear Functions: Finding Zeros
Form: $f(x) = mx + b$
Process: Set $mx + b = 0$ → Solve for $x$ → $x = -\dfrac{b}{m}$
A linear function (with $m \neq 0$) has exactly one zero.
Worked Examples
A) $f(x) = 3x - 12$
$3x - 12 = 0 \;\Rightarrow\; 3x = 12 \;\Rightarrow\; \boxed{x = 4}$
$3x - 12 = 0 \;\Rightarrow\; 3x = 12 \;\Rightarrow\; \boxed{x = 4}$
B) $f(x) = -2x + 7$
$-2x + 7 = 0 \;\Rightarrow\; 2x = 7 \;\Rightarrow\; \boxed{x = 3.5}$
$-2x + 7 = 0 \;\Rightarrow\; 2x = 7 \;\Rightarrow\; \boxed{x = 3.5}$
C) $f(x) = 5x$
$5x = 0 \;\Rightarrow\; \boxed{x = 0}$ (passes through origin)
$5x = 0 \;\Rightarrow\; \boxed{x = 0}$ (passes through origin)
Special Cases:
• $f(x) = 5$ → No zeros (horizontal line never touches $x$-axis)
• $f(x) = 0$ → Infinitely many zeros (the line IS the $x$-axis)
• $f(x) = 5$ → No zeros (horizontal line never touches $x$-axis)
• $f(x) = 0$ → Infinitely many zeros (the line IS the $x$-axis)
Interactive visualization — available in the full lesson