Question
Evaluate the piecewise function when \( x=0 \) and \( x=6 \) \( f(x)=\left\{\begin{array}{l}3 x+5 \text {, if } x \leq 5 \\ \frac{1}{3} x+2 \text {, if } x>5\end{array}\right. \) The value of \( f \) is \( \square \) when \( x=0 \). The value of \( f \) is \( \square \) when \( x=6 \).
Ask by Parsons Young. in the United States
Jan 24,2025
Real Tutor Solution
Tutor-Verified Answer
Answer
When \( x = 0 \), \( f(x) = 5 \). When \( x = 6 \), \( f(x) = 4 \).
Solution
To evaluate the piecewise function \( f(x) \) at specific values of \( x \), we'll use the appropriate expression based on the given conditions.
The piecewise function is defined as:
\[
f(x) = \begin{cases}
3x + 5 & \text{if } x \leq 5 \\
\frac{1}{3}x + 2 & \text{if } x > 5
\end{cases}
\]
### 1. Evaluating \( f(0) \)
**Step 1:** Determine which condition \( x = 0 \) satisfies.
Since \( 0 \leq 5 \), we use the first expression:
\[
f(x) = 3x + 5
\]
**Step 2:** Substitute \( x = 0 \) into the expression:
\[
f(0) = 3(0) + 5 = 0 + 5 = 5
\]
**Result:**
\[
\boxed{5}
\]
### 2. Evaluating \( f(6) \)
**Step 1:** Determine which condition \( x = 6 \) satisfies.
Since \( 6 > 5 \), we use the second expression:
\[
f(x) = \frac{1}{3}x + 2
\]
**Step 2:** Substitute \( x = 6 \) into the expression:
\[
f(6) = \frac{1}{3}(6) + 2 = 2 + 2 = 4
\]
**Result:**
\[
\boxed{4}
\]
### Final Answers
- The value of \( f \) is \( \boxed{5} \) when \( x = 0 \).
- The value of \( f \) is \( \boxed{4} \) when \( x = 6 \).
Reviewed and approved by the UpStudy tutoring team
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Simplify this solution Mind Expander
To evaluate the piecewise function at \( x=0 \), we note that since \( 0 \leq 5 \), we use the first part of the function: \[ f(0) = 3(0) + 5 = 5. \] Thus, the value of \( f \) is \( 5 \) when \( x=0 \). For \( x=6 \), since \( 6 > 5 \), we use the second part of the function: \[ f(6) = \frac{1}{3}(6) + 2 = 2 + 2 = 4. \] So, the value of \( f \) is \( 4 \) when \( x=6 \). Putting it all together: The value of \( f \) is \( 5 \) when \( x=0 \). The value of \( f \) is \( 4 \) when \( x=6 \).
