ACT Math Functions: Evaluate and Compose Functions Confidently

A function f(x) is a machine: input x, get output f(x). To evaluate, substitute the input value for x. Example: If f(x)=2x+3 and you want f(5), substitute: f(5)=2(5)+3=10+3=13. Composition: f(g(x)) means apply g first, then apply f to the result. Example: f(x)=2x+3, g(x)=x-1. f(g(x))=f(x-1)=2(x-1)+3=2x-2+3=2x+1. Composition is mechanical: wherever you see x in f, replace it with the entire g(x) expression.

Another composition: f(g(5))=f(5-1)=f(4)=2(4)+3=11. First evaluate g(5)=4, then evaluate f(4)=11.

Mistake 1: Confusing f(x) with f times x. f(x) is function notation; it doesn't mean multiply. Mistake 2: Substituting incorrectly in composition. Replace the entire inner function, not just the variable. Mistake 3: Forgetting order of operations when substituting. If f(x)=2x+3 and x=5-1, compute 5-1=4 first, then f(4), not f(5-1) by plugging in directly. Always substitute carefully and follow order of operations.

During practice, write out substitution step-by-step. Don't try to do it in your head; errors happen when you skip steps.

Problem 1: f(x)=3x-2. Evaluate f(4). f(4)=3(4)-2=12-2=10. Problem 2: g(x)=x^2+1. Evaluate g(3). g(3)=(3)^2+1=9+1=10. Problem 3: f(x)=2x+1, g(x)=x-3. Evaluate f(g(2)). g(2)=2-3=-1. f(-1)=2(-1)+1=-2+1=-1. Problem 4: f(x)=x+5, g(x)=2x. Find f(g(x)). f(g(x))=f(2x)=(2x)+5=2x+5. Problem 5: h(x)=(1/2)x+4. Evaluate h(10). h(10)=(1/2)(10)+4=5+4=9. Evaluate and compose each, showing all substitution steps.

Find 10 function problems from a practice test. For each, evaluate or compose carefully, writing out every step. By the tenth problem, function mechanics will be automatic.

Function questions appear on most ACT Math tests, usually in questions 35-50. They test substitution and composition mechanics. Students who master function evaluation and composition pick up 1-2 points because the skills are mechanical and errors are preventable.

Drill functions daily this week. Each day, evaluate and compose five functions. By test day, you should evaluate or compose any function accurately within 45 seconds.