Math Problem Statement
Consider the following piecewise-defined function.
f(x)=⎧⎩⎨⎪⎪⎪⎪−2x−22x2+2x34x−1ifx≤−5if−5<x≤5ifx>5f(x)={−2x−2ifx≤−52x2+2xif−5<x≤534x−1ifx>5
Step 2 of 3 :
Evaluate this function at x=−5x=−5. Express your answer as an integer or simplified fraction. If the function is undefined at the given value, indicate "Undefined".
Solution
To evaluate the piecewise-defined function at , we need to determine which piece of the function applies for .
The given function is:
\begin{cases} -2x - 2, & \text{if } x \leq -5 \\ 2x^2 + 2x, & \text{if } -5 < x \leq 5 \\ 34x - 1, & \text{if } x > 5 \end{cases}$$ Since $$x = -5$$, we use the first piece of the function, $$f(x) = -2x - 2$$. Now, substitute $$x = -5$$ into the expression: $$f(-5) = -2(-5) - 2 = 10 - 2 = 8.$$ Thus, the value of the function at $$x = -5$$ is $$8$$. Would you like more details or have any questions? Here are five related questions to explore further: 1. What is the value of $$f(x)$$ at $$x = 0$$? 2. How would you evaluate $$f(x)$$ at $$x = 6$$? 3. What is the graph of this piecewise function? 4. How can we determine if a piecewise function is continuous at a given point? 5. What happens to the function when $$x$$ is near the boundary points? **Tip:** When evaluating piecewise functions, always check which interval your input belongs to in order to use the correct expression.Ask a new question for Free
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Math Problem Analysis
Mathematical Concepts
Piecewise Functions
Function Evaluation
Formulas
f(x) = -2x - 2 for x ≤ -5
f(x) = 2x^2 + 2x for -5 < x ≤ 5
f(x) = 34x - 1 for x > 5
Theorems
Piecewise Function Evaluation
Suitable Grade Level
Grades 10-12