Math Problem Statement
The piecewise function f(x) is defined as follows:
f(x) =
{
x^2 - 1 if x <= 0,
2x - 1 if 0 < x <= 5,
3 if x > 5
}
Determine if it is a function. Additionally, evaluate f(-2), f(0), and f(5).
Solution
The piecewise function is defined as follows:
\begin{cases} x^2 - 1 & \text{if } x \leq 0 \\ 2x - 1 & \text{if } 0 < x \leq 5 \\ 3 & \text{if } x > 5 \end{cases}$$ We are tasked with answering the following: ### 1. Is $$f(x)$$ a function? To determine whether this is a function, we check if each input $$x$$ gives exactly one output $$f(x)$$. Since for each range of $$x$$, the function gives a single value, $$f(x)$$ **is a function**. ### 2. Evaluate $$f(-2)$$, $$f(0)$$, and $$f(5)$$ - For $$f(-2)$$: Since $$-2 \leq 0$$, use the first piece: $$f(-2) = (-2)^2 - 1 = 4 - 1 = 3$$ - For $$f(0)$$: Since $$0 \leq 0$$, use the first piece again: $$f(0) = (0)^2 - 1 = 0 - 1 = -1$$ - For $$f(5)$$: Since $$0 < 5 \leq 5$$, use the second piece: $$f(5) = 2(5) - 1 = 10 - 1 = 9$$ ### Summary: - $$f(-2) = 3$$ - $$f(0) = -1$$ - $$f(5) = 9$$ Would you like further clarification, or help graphing the function? Here are some related questions: 1. How would you plot the piecewise function on the given grid? 2. What happens to $$f(x)$$ for values like $$x = 6$$ or $$x = -3$$? 3. How do you find the domain and range of $$f(x)$$? 4. What is the continuity of $$f(x)$$ at $$x = 0$$ and $$x = 5$$? 5. How would the function change if the second piece was $$2x + 1$$ instead of $$2x - 1$$? **Tip:** When evaluating piecewise functions, always check which condition the input $$x$$ satisfies to choose the correct expression.Ask a new question for Free
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Math Problem Analysis
Mathematical Concepts
Piecewise Functions
Function Evaluation
Algebra
Formulas
f(x) = x^2 - 1 for x <= 0
f(x) = 2x - 1 for 0 < x <= 5
f(x) = 3 for x > 5
Theorems
Definition of a Function
Suitable Grade Level
Grades 9-11