Math Problem Statement
If f(x) is defined as follows, find (a) f(-4), (b) f(0), and (c) f(2). f(x) is defined as: f(x) = {x^2 if x < 0, 3 if x = 0, 2x + 4 if x > 0}.
Solution
The function is defined piecewise as follows:
\begin{cases} x^2 & \text{if } x < 0 \\ 3 & \text{if } x = 0 \\ 2x + 4 & \text{if } x > 0 \end{cases}$$ Now, we will evaluate the given points: ### (a) $$f(-4)$$: Since $$-4 < 0$$, we use the rule $$f(x) = x^2$$. Therefore: $$f(-4) = (-4)^2 = 16$$ ### (b) $$f(0)$$: Since $$x = 0$$, we use the rule $$f(x) = 3$$. Therefore: $$f(0) = 3$$ ### (c) $$f(2)$$: Since $$2 > 0$$, we use the rule $$f(x) = 2x + 4$$. Therefore: $$f(2) = 2(2) + 4 = 4 + 4 = 8$$ Thus, the answers are: - $$f(-4) = 16$$ - $$f(0) = 3$$ - $$f(2) = 8$$ Would you like more details on any part of the solution? Here are 5 related questions to explore: 1. How would the value of $$f(x)$$ change if $$f(x) = x^3$$ for $$x < 0$$? 2. What is $$f(1.5)$$ for the same function definition? 3. How do you determine whether to use $$f(x) = x^2$$ or $$f(x) = 2x + 4$$ for other negative or positive values? 4. Can you graph the piecewise function to visualize its behavior for all values of $$x$$? 5. What happens to $$f(x)$$ as $$x$$ approaches infinity or negative infinity? **Tip**: In piecewise functions, always check which condition applies to the given value of $$x$$ before plugging it into the function.Ask a new question for Free
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Math Problem Analysis
Mathematical Concepts
Piecewise Functions
Function Evaluation
Formulas
f(x) = x^2 for x < 0
f(x) = 3 for x = 0
f(x) = 2x + 4 for x > 0
Theorems
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Suitable Grade Level
Grades 9-11