KKhaemenes PrecalculusTeacher Answer Key
Student Lesson

Teacher Key · Unit 01 · Lesson 04

Transformations and Families of Functions

Use this key to verify method, notation, domain, units, and interpretation. Accept equivalent exact forms when mathematically valid.

Khaemenes Academy · Teacher Key

Transformations and Families of Functions

U01 · L04
1

For f(x)=2x²−3 and x=6, evaluate f(x).

Functions visualAnalyze functions through algebraic, graphical, numerical, verbal, and contextual representations, then defend what each representation reveals. xy
Answer: 69
Reasoning: Substitute x=6: 2(6)²−3=69.
2

State the domain of g(x)=√(x−4).

Functions visualAnalyze functions through algebraic, graphical, numerical, verbal, and contextual representations, then defend what each representation reveals. xy
Answer: [4,∞)
Reasoning: The radicand must be nonnegative: x−4≥0.
3

Find the average rate of change of f(x)=x² from x=1 to x=4.

Functions visualAnalyze functions through algebraic, graphical, numerical, verbal, and contextual representations, then defend what each representation reveals. xy
Answer: 5
Reasoning: [16−1]/[4−1]=15/3=5.
4

Describe the transformation from y=x² to y=−2(x−3)²+1.

Functions visualAnalyze functions through algebraic, graphical, numerical, verbal, and contextual representations, then defend what each representation reveals. xy
Answer: Right 3, reflect across x-axis, vertical stretch by 2, up 1
Reasoning: Read the form a(x−h)²+k.
5

If f(x)=x+2 and g(x)=3x, find (f∘g)(4).

Functions visualAnalyze functions through algebraic, graphical, numerical, verbal, and contextual representations, then defend what each representation reveals. xy
Answer: 14
Reasoning: g(4)=12 and f(12)=14.
6

Find the inverse of f(x)=5x−7.

Functions visualAnalyze functions through algebraic, graphical, numerical, verbal, and contextual representations, then defend what each representation reveals. xy
Answer: f⁻¹(x)=(x+7)/5
Reasoning: Swap x and y and solve for y.
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