Math Problem Statement

Let f(x) = { x if x ≥ 5, 5 if x < 5 }. Find f(3), f(5), and f(7). Sketch the graph of y = f(x).

Solution

The problem presents a piecewise function:

\begin{cases} x & \text{if } x \geq 5 \\ 5 & \text{if } x < 5 \end{cases}$$ ### (a) Evaluate $$f(3)$$, $$f(5)$$, and $$f(7)$$: 1. **For $$f(3)$$:** Since $$3 < 5$$, we use $$f(x) = 5$$. Therefore, $$f(3) = 5$$. 2. **For $$f(5)$$:** Since $$5 \geq 5$$, we use $$f(x) = x$$. Therefore, $$f(5) = 5$$. 3. **For $$f(7)$$:** Since $$7 \geq 5$$, we use $$f(x) = x$$. Therefore, $$f(7) = 7$$. ### Final answers: - $$f(3) = 5$$ - $$f(5) = 5$$ - $$f(7) = 7$$ --- ### (b) Sketch the graph of $$y = f(x)$$: 1. For $$x \geq 5$$, the graph is a line $$y = x$$, starting at the point $$(5, 5)$$ and increasing. 2. For $$x < 5$$, the graph is a horizontal line $$y = 5$$, for all $$x < 5$$. The graph has a discontinuity (a jump) at $$x = 5$$, where $$f(5) = 5$$ (from the $$y = x$$ line). Let me know if you'd like further explanations or assistance sketching this! --- ### Related Questions: 1. What is the domain and range of $$f(x)$$? 2. Is $$f(x)$$ continuous? Why or why not? 3. What type of function is $$f(x)$$ for $$x \geq 5$$ and $$x < 5$$? 4. How would the graph of $$f(x)$$ change if the conditions were reversed? 5. What is the slope of the $$y = x$$ portion of $$f(x)$$? --- **Tip:** For piecewise functions, always check which part of the definition applies to each $$x$$ value!

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Math Problem Analysis

Mathematical Concepts

Piecewise Functions
Graphing Functions
Function Evaluation

Formulas

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Theorems

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Suitable Grade Level

Grades 9-12