Math Problem Statement

Can you graph the function f and then solve the question

Can you graph the function f and then solve the question?

Solution

We have successfully graphed the function f(x)f(x), which is a piecewise function defined as:

\begin{cases} 8 - x^2 & \text{for } x < 5 \\ x + 8 & \text{for } x \geq 5 \end{cases}$$ ### Step-by-Step Solution: We are asked to find the one-sided limit $$\lim_{x \to 5} g(f(x))$$. To proceed, let’s analyze $$f(x)$$ as $$x \to 5$$: - **From the left**: For $$x < 5$$, $$f(x) = 8 - x^2$$. At $$x = 5$$, this yields $$f(5^-) = 8 - 25 = -17$$. - **From the right**: For $$x \geq 5$$, $$f(x) = x + 8$$. At $$x = 5$$, this gives $$f(5^+) = 5 + 8 = 13$$. Thus, we need to compute the limits $$\lim_{x \to -17^+} g(x)$$ and $$\lim_{x \to 13^-} g(x)$$ based on the behavior of $$f(x)$$ as $$x$$ approaches 5. Since we are dealing with one-sided limits, it seems that the answer will be linked to options involving $$\lim_{x \to 13^-} g(x)$$ or $$\lim_{x \to -17^+} g(x)$$. The most appropriate option is likely **C: $$\lim_{x \to -17^+} g(x)$$**. Would you like more details or clarification? Here are some related questions to expand on this concept: 1. How do you evaluate limits of composite functions like $$g(f(x))$$? 2. What is the difference between one-sided and two-sided limits? 3. Can you explain the behavior of piecewise functions at their breaking points? 4. What role does continuity play in the evaluation of limits? 5. How can we graphically verify the existence of limits? **Tip:** Always analyze one-sided limits by checking the behavior from both directions separately!

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Math Problem Analysis

Mathematical Concepts

Piecewise Functions
Limits
One-Sided Limits
Function Composition

Formulas

f(x) = 8 - x^2 for x < 5
f(x) = x + 8 for x ≥ 5
lim_{x→5} g(f(x))

Theorems

Limits of Piecewise Functions
One-Sided Limits

Suitable Grade Level

Grades 11-12 or Early College

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