Answer:
(a) -1
(b) 4
(c) 14
(d) 61
Step-by-step explanation:
A piecewise function is a function which has different definitions for different intervals of x.
Given:
[tex]f(x)=\begin{cases}5x-1 \quad \textsf{if }-5\leq x \leq 3\\x^3-3 \quad \textsf{if }\:\:\:\:\:\:3 < x \leq 4\end{cases}[/tex]
f(x) has 2 definitions:
Definition 1
[tex]5x-1[/tex] when x is more than or equal to -5 and less than or equal to 3. This is a linear function.
Definition 2
[tex]x^2-3[/tex] when x is more than 3 and less than or equal to 4. This is a cubic function.
Part (a)
We have to find f(0), so when x = 0.
x = 0 satisfies the condition -5 ≤ x ≤ 3 so the corresponding function is
[tex]f(x)=5x-1[/tex]
Substitute x = 0 in this definition:
[tex]\implies f(0)=5(0)-1=-1[/tex]
Part (b)
We have to find f(1), so when x = 1.
x = 1 satisfies the condition -5 ≤ x ≤ 3 so the corresponding function is
[tex]f(x)=5x-1[/tex]
Substitute x = 1 in this definition:
[tex]\implies f(0)=5(1)-1=4[/tex]
Part (c)
We have to find f(3), so when x = 3.
x = 3 satisfies the condition -5 ≤ x ≤ 3 so the corresponding function is
[tex]f(x)=5x-1[/tex]
Substitute x = 3 in this definition:
[tex]\implies f(0)=5(3)-1=14[/tex]
Part (d)
We have to find f(4), so when x = 4.
x = 4 satisfies the condition 3 < x ≤ 4 so the corresponding function is
[tex]f(x)=x^3-3[/tex]
Substitute x = 4 in this definition:
[tex]\implies f(4)=(4)^3-3=61[/tex]
