Question

if $f(x) = \

$$\begin{cases} 5x - 2 & \\text{if } -5 \\leq x \\leq 3 \\\\ x^3 - 3 & \\text{if } 3 < x \\leq 4 \\end{cases}$$

$, find: (a) $f(0)$, (b) $f(1)$, (c) $f(3)$, and (d) $f(4)$.

(a) $f(0) = \square$

(b) $f(1) = \square$

(c) $f(3) = \square$

(d) $f(4) = \square$

Explanation:

Step1: Determine the function for \( f(0) \)

Since \( -5 \leq 0 \leq 3 \), we use \( f(x) = 5x - 2 \). Substitute \( x = 0 \): \( f(0) = 5(0) - 2 = -2 \).

Step2: Determine the function for \( f(1) \)

Since \( -5 \leq 1 \leq 3 \), we use \( f(x) = 5x - 2 \). Substitute \( x = 1 \): \( f(1) = 5(1) - 2 = 3 \).

Step3: Determine the function for \( f(3) \)

Since \( -5 \leq 3 \leq 3 \) (the first interval includes 3), we use \( f(x) = 5x - 2 \). Substitute \( x = 3 \): \( f(3) = 5(3) - 2 = 13 \).

Step4: Determine the function for \( f(4) \)

Since \( 3 < 4 \leq 4 \), we use \( f(x) = x^3 - 3 \). Substitute \( x = 4 \): \( f(4) = 4^3 - 3 = 64 - 3 = 61 \).

Answer:

(a) \( f(0) = -2 \)
(b) \( f(1) = 3 \)
(c) \( f(3) = 13 \)
(d) \( f(4) = 61 \)