Question
Evaluate the piecewise function at the given values of the independent variable. \( f(x)=\left\{\begin{array}{ll}5 x+4 & \text { if } x<0 \\ 2 x+5 & \text { if } x \geq 0\end{array}\right. \) (a) \( f(-1) \) (b) \( f(0) \) (c) \( f(2) \)
Ask by Elliott Brewer. in the United States
Jan 25,2025
Real Tutor Solution
Tutor-Verified Answer
Answer
\( f(-1) = -1 \), \( f(0) = 5 \), \( f(2) = 9 \).
Solution
Let's evaluate the piecewise function \( f(x) \) at the specified values of \( x \):
\[
f(x) = \begin{cases}
5x + 4 & \text{if } x < 0 \\
2x + 5 & \text{if } x \geq 0
\end{cases}
\]
### (a) \( f(-1) \)
**Step 1:** Determine which part of the piecewise function to use.
Since \( -1 < 0 \), we use:
\[
f(x) = 5x + 4
\]
**Step 2:** Substitute \( x = -1 \) into the function.
\[
f(-1) = 5(-1) + 4 = -5 + 4 = -1
\]
**Answer:** \( f(-1) = -1 \)
---
### (b) \( f(0) \)
**Step 1:** Determine which part of the piecewise function to use.
Since \( 0 \geq 0 \), we use:
\[
f(x) = 2x + 5
\]
**Step 2:** Substitute \( x = 0 \) into the function.
\[
f(0) = 2(0) + 5 = 0 + 5 = 5
\]
**Answer:** \( f(0) = 5 \)
---
### (c) \( f(2) \)
**Step 1:** Determine which part of the piecewise function to use.
Since \( 2 \geq 0 \), we use:
\[
f(x) = 2x + 5
\]
**Step 2:** Substitute \( x = 2 \) into the function.
\[
f(2) = 2(2) + 5 = 4 + 5 = 9
\]
**Answer:** \( f(2) = 9 \)
---
**Summary of Results:**
- **(a)** \( f(-1) = -1 \)
- **(b)** \( f(0) = 5 \)
- **(c)** \( f(2) = 9 \)
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Simplify this solution The Deep Dive
Let's evaluate the piecewise function \( f(x) \) at the given values. (a) For \( f(-1) \): Since \(-1 < 0\), we use the first part of the function: \[ f(-1) = 5(-1) + 4 = -5 + 4 = -1. \] (b) For \( f(0) \): Since \(0 \geq 0\), we use the second part of the function: \[ f(0) = 2(0) + 5 = 0 + 5 = 5. \] (c) For \( f(2) \): Since \(2 \geq 0\), we use the second part of the function: \[ f(2) = 2(2) + 5 = 4 + 5 = 9. \] So the evaluations are: (a) \( f(-1) = -1 \) (b) \( f(0) = 5 \) (c) \( f(2) = 9 \)
