Question

sketch a graph of the following equations. after graphing, state the domain and range and answer questions a, b, and c.

1. $f(x) = \

$$\begin{cases} 2x + 5 & \\text{if } -3 \\leq x < 0 \\\\ -3 & \\text{if } x = 0 \\\\ -5x & \\text{if } x > 0 \\end{cases}$$

$
find a) $f(-2)$ \quad b) $f(0)$ \quad c) $f(3)$

1. $f(x) = \

$$\begin{cases} -x + 1 & \\text{if } -2 \\leq x < 1 \\\\ 4 & \\text{if } x = 1 \\\\ x^2 & \\text{if } x > 1 \\end{cases}$$

$
find a) $f(-1)$ \quad b) $f(1)$ \quad c) $f(2)$

Explanation:

Problem 1:

a) Find \( f(-2) \)

Step1: Determine the domain for \( x = -2 \)
Since \( -3 \leq -2 < 0 \), we use the first part of the piecewise function: \( f(x) = 2x + 5 \)
Step2: Substitute \( x = -2 \) into \( 2x + 5 \)
\( f(-2) = 2(-2) + 5 = -4 + 5 = 1 \)

b) Find \( f(0) \)

Step1: Determine the domain for \( x = 0 \)
Since \( x = 0 \), we use the second part of the piecewise function: \( f(x) = -3 \)
Step2: Substitute \( x = 0 \) into \( -3 \)
\( f(0) = -3 \)

c) Find \( f(3) \)

Step1: Determine the domain for \( x = 3 \)
Since \( 3 > 0 \), we use the third part of the piecewise function: \( f(x) = -5x \)
Step2: Substitute \( x = 3 \) into \( -5x \)
\( f(3) = -5(3) = -15 \)

Problem 2:

a) Find \( f(-1) \)

Step1: Determine the domain for \( x = -1 \)
Since \( -2 \leq -1 < 1 \), we use the first part of the piecewise function: \( f(x) = -x + 1 \)
Step2: Substitute \( x = -1 \) into \( -x + 1 \)
\( f(-1) = -(-1) + 1 = 1 + 1 = 2 \)

b) Find \( f(1) \)

Step1: Determine the domain for \( x = 1 \)
Since \( x = 1 \), we use the second part of the piecewise function: \( f(x) = 4 \)
Step2: Substitute \( x = 1 \) into \( 4 \)
\( f(1) = 4 \)

c) Find \( f(2) \)

Step1: Determine the domain for \( x = 2 \)
Since \( 2 > 1 \), we use the third part of the piecewise function: \( f(x) = x^2 \)
Step2: Substitute \( x = 2 \) into \( x^2 \)
\( f(2) = 2^2 = 4 \)

Answer:

Problem 1:

a) \( f(-2) = 1 \)
b) \( f(0) = -3 \)
c) \( f(3) = -15 \)

Problem 2:

a) \( f(-1) = 2 \)
b) \( f(1) = 4 \)
c) \( f(2) = 4 \)