Math Problem Statement
Solution
The problem involves a piecewise function that is defined in three intervals, and we are tasked with finding the range of the function. Let's go through the function step by step:
Function Definition:
\begin{cases} x + 1 & \text{if } x \leq -1 \\ x^2 - 1 & \text{if } -1 < x \leq 2 \\ -3x + 9 & \text{if } x > 2 \end{cases}$$ ### Analyzing Each Piece: 1. **For $$x \leq -1$$,** we have $$f(x) = x + 1$$: - The smallest value of $$x$$ in this interval is $$x = -1$$, and at this point: $$f(-1) = -1 + 1 = 0$$ - As $$x \to -\infty$$, $$f(x) \to -\infty$$. - So, for $$x \leq -1$$, $$f(x)$$ takes values from $$(-\infty, 0]$$. 2. **For $$-1 < x \leq 2$$,** we have $$f(x) = x^2 - 1$$: - This is a quadratic function that opens upwards, with the vertex at $$x = 0$$: $$f(0) = 0^2 - 1 = -1$$ - The function increases as $$x$$ moves away from 0. At $$x = 2$$: $$f(2) = 2^2 - 1 = 3$$ - So, for $$-1 < x \leq 2$$, $$f(x)$$ takes values from $$[-1, 3]$$. 3. **For $$x > 2$$,** we have $$f(x) = -3x + 9$$: - This is a linear function with a negative slope, decreasing as $$x$$ increases. At $$x = 2$$, this function would give: $$f(2) = -3(2) + 9 = 3$$ but this value is already included in the second piece. - As $$x \to \infty$$, $$f(x) \to -\infty$$. - So, for $$x > 2$$, $$f(x)$$ takes values from $$(-\infty, 3]$$. ### Range of the Function: - From the analysis of all three intervals, the range of $$f(x)$$ combines the intervals: - $$(-\infty, 0]$$ from the first piece, - $$[-1, 3]$$ from the second piece, - and $$(-\infty, 3]$$ from the third piece. Therefore, the range of the function is $$(-\infty, 3]$$. ### Correct Answer: The correct option is **e. $$\{ y \mid y \leq 3 \}$$**. --- Do you need further explanation or clarifications? Here are some related questions for practice: 1. What is the domain of the function $$f(x)$$? 2. How would the range change if the middle interval was $$-1 < x \leq 3$$? 3. Can you sketch the graph of $$f(x)$$? 4. What happens to the range if we add a constant to each piece of the function? 5. How do you determine the domain of a piecewise function? **Tip:** Always examine the behavior of the function at boundary points when determining the range of a piecewise function.Ask a new question for Free
By Image
Drop file here or Click Here to upload
Math Problem Analysis
Mathematical Concepts
Piecewise Functions
Function Range
Linear and Quadratic Functions
Formulas
f(x) = x+1 untuk x ≤ -1
f(x) = x^2 - 1 untuk -1 < x ≤ 2
f(x) = -3x + 9 untuk x > 2
Theorems
Properties of Quadratic Functions
Linear Function Behavior
Range Determination for Piecewise Functions
Suitable Grade Level
Grades 10-12
Related Recommendation
Domain and Range of a Piecewise Function with Linear Segments
Finding the Range of a Piecewise Function with Two Linear Expressions
Graphing and Finding the Range of a Piecewise Function
Determine the Range of a Piecewise-Defined Function Given a Specific Domain
Determine Domain and Range of a Piecewise Function and Write the Equation