ACT Math: Evaluate and Compose Functions Without Confusion

Function notation f(x) means "the output when input is x." To evaluate: (1) Identify the function rule: f(x)=2x+3. (2) Substitute the input value for every x. If asked for f(5), substitute 5: f(5)=2(5)+3=13. If asked for f(-2), substitute -2: f(-2)=2(-2)+3=-1. The key is replacing every instance of x with the given value, including in exponents and denominators.

More complex example: f(x)=x^2-4x+1. Find f(3). f(3)=3^2-4(3)+1=9-12+1=-2. Another: f(x)=1/x. Find f(1/2). f(1/2)=1/(1/2)=2. Notice how substitution works even with fractions and exponents; the rule is mechanically consistent.

Composition f(g(x)) means "apply g first, then apply f to the result." Work from the inside out: (1) Evaluate g(x) at the given input. (2) Take that result and plug it into f. Example: f(x)=2x+1, g(x)=x^2. Find f(g(3)). Step 1: g(3)=3^2=9. Step 2: f(9)=2(9)+1=19. Another example: Find g(f(3)). Step 1: f(3)=2(3)+1=7. Step 2: g(7)=7^2=49. Notice f(g(3))≠g(f(3)), so order matters. Write intermediate results clearly; do not try to compose in your head; errors happen fast when you skip steps.

Practice: f(x)=x+5, g(x)=3x. Find f(g(2)). g(2)=3(2)=6. f(6)=6+5=11. Find g(f(2)). f(2)=2+5=7. g(7)=3(7)=21. Different answers because composition is not commutative.

f(x)=x^2+2x, g(x)=x-3, h(x)=1/x. Evaluate: 1) f(4). 2) g(-1). 3) h(2). 4) f(g(5)). 5) g(f(2)). 6) h(g(1)). For each, show your substitution step-by-step. Do this twice this week; function evaluation speed increases dramatically with repetition.

Answers: 1) f(4)=4^2+2(4)=16+8=24. 2) g(-1)=-1-3=-4. 3) h(2)=1/2. 4) g(5)=5-3=2, f(2)=2^2+2(2)=8. 5) f(2)=2^2+2(2)=8, g(8)=8-3=5. 6) g(1)=1-3=-2, h(-2)=1/(-2)=-1/2.

Function evaluation and composition are foundational for more advanced topics like inverse functions and transformations. Mastering substitution and composition takes 30 minutes of drilling and unlocks 1-2 additional points on harder problems because you can now confidently evaluate any function structure.

Commit to the six-problem drill twice this week. By test day, you will evaluate functions and compositions instantly, without hesitation.