Math Problem Statement
=⎧⎩⎨⎪⎪1−𝑥,2,𝑥+1, if 𝑥≤−1, if −1<𝑥<1, if 𝑥≥1.
a. 𝑓(−2)=
2 functionsequation editor Help on numbers.
b. 𝑓(1)=
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c. For what values of 𝑥 is 𝑓(𝑥)=2 ?
functionsequation editor Help on intervals.
d. Find the domain and range of 𝑓 . (You may find it helpful to graph this function on your own paper to find the domain and range.)
Domain:
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Range:
functionsequation editor Help on intervals.
Solution
The given function can be interpreted as a piecewise function, which is written as:
\begin{cases} 1 - x, & \text{if } x \leq -1 \\ 2, & \text{if } -1 < x < 1 \\ x + 1, & \text{if } x \geq 1 \end{cases}$$ Let's break down and solve each part of the question. --- **a. $$f(-2)$$** To find $$f(-2)$$, we need to check which piece of the function applies for $$x = -2$$. Since $$-2 \leq -1$$, we use the first part of the function $$f(x) = 1 - x$$. $$f(-2) = 1 - (-2) = 1 + 2 = 3$$ So, $$f(-2) = 3$$. --- **b. $$f(1)$$** Now, let's find $$f(1)$$. Since $$1 \geq 1$$, we use the third part of the function, $$f(x) = x + 1$$. $$f(1) = 1 + 1 = 2$$ So, $$f(1) = 2$$. --- **c. For what values of $$x$$ is $$f(x) = 2$$?** We need to solve $$f(x) = 2$$ for each piece of the function. 1. For the first piece, $$f(x) = 1 - x$$, we solve $$1 - x = 2$$: $$1 - x = 2 \implies -x = 1 \implies x = -1$$ So, $$x = -1$$. 2. For the second piece, $$f(x) = 2$$ is already given, so for $$-1 < x < 1$$, $$f(x) = 2$$ for all $$x$$ in this interval. 3. For the third piece, $$f(x) = x + 1$$, we solve $$x + 1 = 2$$: $$x + 1 = 2 \implies x = 1$$ So, $$x = 1$$. Therefore, $$f(x) = 2$$ for $$x \in (-1, 1]$$. --- **d. Find the domain and range of $$f$$** - **Domain**: The function is defined for all real numbers $$x$$. Therefore, the domain is: $$\text{Domain} = (-\infty, \infty)$$ - **Range**: The function outputs different values depending on the intervals: - For $$x \leq -1$$, $$f(x) = 1 - x$$, which is a line with decreasing values. - For $$-1 < x < 1$$, $$f(x) = 2$$. - For $$x \geq 1$$, $$f(x) = x + 1$$, which increases without bound. Therefore, the range is: $$\text{Range} = [2, \infty)$$ --- Would you like further details or clarifications? --- **5 Related Questions:** 1. What is the behavior of $$f(x)$$ as $$x \to -\infty$$? 2. How would the graph of this piecewise function look? 3. For what values of $$x$$ is $$f(x) = 3$$? 4. How does the range change if one of the intervals in the piecewise function is modified? 5. What is the significance of continuity in this piecewise function? **Tip**: When working with piecewise functions, always check which condition applies to the specific $$x$$-value before solving!Ask a new question for Free
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Math Problem Analysis
Mathematical Concepts
Piecewise Functions
Domain and Range
Solving Equations
Formulas
f(x) = 1 - x for x ≤ -1
f(x) = 2 for -1 < x < 1
f(x) = x + 1 for x ≥ 1
Theorems
-
Suitable Grade Level
Grades 9-12
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