Question
TRY IT! \[ \left.\begin{array}{lr}f(x)=\left\{\begin{array}{lr}2 x^{3}-1, & x<1 \\ 3, & 1 \leq x<5 \\ |x-2|, & x \geq 5\end{array}\right. \\ f(8)=60 & f(0)=-1\end{array} \right\rvert\, f(4)= \]
Ask by Mitchell Murray. in the United States
Feb 03,2025
Real Tutor Solution
Tutor-Verified Answer
Answer
\( f(4) = 3 \)
Solution
To find \( f(4) \) using the piecewise function defined as:
\[
f(x) =
\begin{cases}
2x^3 - 1 & \text{if } x < 1 \\
3 & \text{if } 1 \leq x < 5 \\
|x - 2| & \text{if } x \geq 5
\end{cases}
\]
we need to determine which condition \( x = 4 \) falls under.
1. **Identify the interval for \( x = 4 \)**:
- The first case \( 2x^3 - 1 \) applies for \( x < 1 \).
- The second case \( 3 \) applies for \( 1 \leq x < 5 \).
- The third case \( |x - 2| \) applies for \( x \geq 5 \).
Since \( 4 \) falls within the interval \( 1 \leq x < 5 \), we use the second case.
2. **Evaluate \( f(4) \)**:
- According to the second case, \( f(4) = 3 \).
Thus, the final answer is:
\[
f(4) = 3
\]
Reviewed and approved by the UpStudy tutoring team
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Simplify this solution Beyond the Answer
To find \( f(4) \), we start by determining which piece of the piecewise function to use for \( x = 4 \). Since \( 1 \leq 4 < 5 \), we can see that we should use the middle piece, which is \( f(x) = 3 \). Therefore, \( f(4) = 3 \). When considering function evaluations like this one, it’s always essential to clearly identify the intervals defined in a piecewise function to avoid errors. A common mistake is to overlook the boundaries, so make sure to check which interval the input lies in before making a calculation!
