Functions and Their Domains in AP Pre-Calculus

AP pre calculus function and domain

Understanding function transformations is one of the most important skills in AP Precalculus and advanced algebra. In this problem, we analyze how a function’s domain changes when it is transformed.

Problem

Let f be a function with the domain −2≤x≤8. The function g is defined by: g(x)=3f(2(x+4))−1.

Which of the following gives the domain of the function g?

(A) [−7,23]
(B) [−5,0]
(C) [−3,2]
(D) [0,20]

Step-by-Step Solution

The key idea is simple but powerful: 👉 The input inside function f must stay within its domain.

We know:

• Domain of f: −2≤x≤8

But in function g(x), the input to f is: 2(x+4). So we require: −2≤2(x+4)≤8

Solve the inequality:

−2≤2x+8≤8

Subtract 8 from all parts: −10≤2x≤0

Divide everything by 2: −5≤x≤0

Final Answer

The domain of g(x) is: [−5,0]

✅ Correct choice: (B)

Why This Matters for AP Precalculus

This type of question tests your understanding of:

• Function composition
• Domain restrictions
• Transformations inside functions
• Inequalities and algebra manipulation

These concepts appear frequently on AP Precalculus exams, SAT Math, and NYC specialized high school tests (SHSAT).

Common Mistakes

• ❌ Forgetting that the restriction applies to the inside of the function
• ❌ Not solving the inequality correctly
• ❌ Ignoring transformations like shifts and stretches

Learn Precalculus the Right Way

At School of Math NYC, we help students master topics like:

• Function transformations
• Domain and range
• AP Precalculus exam strategies
• SHSAT and advanced math preparation
• AP pre calculus function and domain

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