Functions are one of the highest-value topics in college algebra because they connect equations, graphs, tables, and real-world models. If a student can find domain and range, evaluate function notation, read graphs, and recognize transformations, the rest of college algebra becomes much easier to organize.
This practice guide gives worked examples first, then a short mixed set with answers.
Function Skill 1: Domain
The domain is the set of input values that are allowed. In college algebra, the most common restrictions come from square roots and denominators.
Example 1
Find the domain of f(x) = sqrt(x + 2).
The expression inside the square root must be nonnegative:
x + 2 >= 0, so x >= -2.
Domain: x >= -2.
Example 2
Find the domain of g(x) = 5 / (x – 4).
The denominator cannot equal 0, so x – 4 cannot equal 0. Therefore x cannot equal 4.
Domain: all real numbers except x = 4.
Function Skill 2: Range
The range is the set of output values. For graph-based questions, read the lowest and highest y-values. For common parent functions, think about the shape.
Example 3
Find the range of f(x) = (x – 1)^2 + 3.
This is a parabola opening upward. Its vertex is (1, 3), so the smallest y-value is 3.
Range: y >= 3.
Function Skill 3: Function Notation
Function notation means substitution. If f(x) = 2x^2 – 5, then f(3) means replace x with 3:
f(3) = 2(3^2) – 5 = 18 – 5 = 13.
For composite functions, work from the inside out.
If f(x) = 2x + 1 and g(x) = x^2, then f(g(4)) = f(16) = 33.
Function Skill 4: Graphs and Transformations
Many college algebra problems ask how a graph changes. Use these rules:
- f(x) + k shifts the graph up k units.
- f(x) – k shifts the graph down k units.
- f(x – h) shifts the graph right h units.
- f(x + h) shifts the graph left h units.
- -f(x) reflects the graph over the x-axis.
- f(-x) reflects the graph over the y-axis.
- a f(x) vertically stretches the graph if |a| > 1 and compresses it if 0 < |a| < 1.
Function Skill 5: Interpreting a Function in Context
College algebra questions often use functions to model cost, height, temperature, revenue, or distance. In those questions, do not treat the formula as abstract symbols only. Identify what the input means, what the output means, and whether a value makes sense in the situation.
For example, if C(n) = 12n + 25 models the cost of n tickets, then n should be a whole number greater than or equal to 0. The number 25 is a fixed fee, and 12 is the cost per ticket. A domain answer like “all real numbers” is mathematically possible for the formula but not reasonable for the context because you cannot buy -3.4 tickets.
Practice Questions
1. Find the domain of f(x) = sqrt(x – 7).
2. Find the domain of g(x) = (x + 2) / (x^2 – 9).
3. Find the range of h(x) = -2(x + 4)^2 + 5.
4. If f(x) = 3x – 8, find f(6).
5. If f(x) = x^2 + 2 and g(x) = x – 5, find f(g(9)).
6. The point (2, 7) is on the graph of f. What point is on the graph of f(x) – 4?
7. The graph of y = sqrt(x) is shifted left 3 and down 2. Write the new equation.
8. A function table includes (0, 4), (1, 7), (2, 10), and (3, 13). Write a linear function.
9. A graph has a vertical asymptote at x = -1. What x-value is excluded from the domain?
10. A function models cost C(n) = 12n + 25. What does 25 represent?
11. Find the domain of p(x) = 1 / (x + 5).
12. Find the domain of q(x) = sqrt(10 – x).
13. If f(x) = 2x – 3 and g(x) = x + 8, find g(f(5)).
14. A parabola opens upward and has vertex (-2, -7). What is its range?
15. The graph of y = |x| is reflected over the x-axis and shifted up 6. Write the new equation.
16. A gym charges M(w) = 18w + 40 for w weeks. What is the meaning of 18, and what practical domain makes sense?
Answer Key
1. x – 7 >= 0, so x >= 7.
2. x^2 – 9 = (x – 3)(x + 3), so x cannot equal 3 or -3.
3. The parabola opens downward and has vertex (-4, 5), so the range is y <= 5.
4. f(6) = 3(6) – 8 = 10.
5. g(9) = 4. Then f(4) = 16 + 2 = 18.
6. f(x) – 4 shifts outputs down 4, so (2, 7) becomes (2, 3).
7. Left 3 means x + 3. Down 2 means subtract 2. Equation: y = sqrt(x + 3) – 2.
8. y increases by 3 each time and y = 4 when x = 0, so y = 3x + 4.
9. x = -1 is excluded.
10. The 25 is the fixed starting cost or fee before any units are added.
11. x + 5 cannot equal 0, so x cannot equal -5.
12. 10 – x >= 0, so x <= 10.
13. f(5) = 2(5) – 3 = 7. Then g(7) = 7 + 8 = 15.
14. The smallest output is -7, so the range is y >= -7.
15. Reflecting over the x-axis gives -|x|. Shifting up 6 gives y = -|x| + 6.
16. The 18 is the weekly charge. A practical domain is whole numbers w >= 0, because weeks of membership are not negative.
Common Mistakes
Solving the Range Like the Domain
Domain restrictions usually come from the input. Range requires thinking about output values. For quadratics, start with the vertex.
Moving Graphs in the Wrong Direction
Inside changes can feel backward: f(x – 3) moves right 3, while f(x + 3) moves left 3. Outside changes are direct: f(x) + 3 moves up 3.
Confusing f(g(x)) with Multiplication
Composition means substitute the inside function into the outside function. It is not f(x) times g(x).
Ignoring Context Restrictions
A formula may allow many inputs that the real situation does not. If x represents people, tickets, weeks, or objects, the practical domain is usually whole numbers and cannot include negative values.
One-Week Functions Review Plan
| Day | Focus | Checkpoint |
|---|---|---|
| 1 | Domain from roots and denominators | Explain every excluded value or inequality. |
| 2 | Range from graphs and vertices | State whether the graph has a lowest or highest point. |
| 3 | Function notation and composition | Write each substitution step before simplifying. |
| 4 | Transformations | Match each equation change to a graph movement. |
| 5 | Mixed application problems | Identify input, output, rate, fixed value, and practical domain. |
Next Study Step
After this practice set, move into mixed college algebra work: equations, inequalities, polynomial functions, rational functions, exponential functions, logarithms, and systems. If functions are still slow, choose a resource with worked examples before timed tests. If the ideas are clear but errors happen under pressure, use mixed problem sets and check every missed item for a domain, substitution, graph-reading, or context error.
Browse ViewMath College Algebra resources for structured practice and review.
