To evaluate the piecewise function at given values of the independent variable, we can follow the conditions specified by the function definition:
[tex]\[
f(x)=\left\{\begin{array}{ll}
5 x+4 & \text { if } x<0 \\
4 x+7 & \text { if } x \geq 0
\end{array}\right.
\][/tex]
Let's evaluate this function step-by-step for each specified value of [tex]\(x\)[/tex].
(a) Evaluate [tex]\(f(-1)\)[/tex]:
Here, [tex]\(x = -1\)[/tex]. According to the piecewise function:
[tex]\[ f(x) = 5x + 4 \quad \text{if } x < 0 \][/tex]
Since [tex]\( -1 < 0 \)[/tex], we use the first part of the function:
[tex]\[ f(-1) = 5(-1) + 4 \][/tex]
[tex]\[ f(-1) = -5 + 4 \][/tex]
[tex]\[ f(-1) = -1 \][/tex]
So, the value of [tex]\( f(-1) \)[/tex] is [tex]\( -1 \)[/tex].
(b) Evaluate [tex]\(f(0)\)[/tex]:
Here, [tex]\(x = 0\)[/tex]. According to the piecewise function:
[tex]\[ f(x) = 4x + 7 \quad \text{if } x \geq 0 \][/tex]
Since [tex]\( 0 \geq 0 \)[/tex], we use the second part of the function:
[tex]\[ f(0) = 4(0) + 7 \][/tex]
[tex]\[ f(0) = 0 + 7 \][/tex]
[tex]\[ f(0) = 7 \][/tex]
So, the value of [tex]\( f(0) \)[/tex] is [tex]\( 7 \)[/tex].
(c) Evaluate [tex]\(f(3)\)[/tex]:
Here, [tex]\(x = 3\)[/tex]. According to the piecewise function:
[tex]\[ f(x) = 4x + 7 \quad \text{if } x \geq 0 \][/tex]
Since [tex]\( 3 \geq 0 \)[/tex], we use the second part of the function:
[tex]\[ f(3) = 4(3) + 7 \][/tex]
[tex]\[ f(3) = 12 + 7 \][/tex]
[tex]\[ f(3) = 19 \][/tex]
So, the value of [tex]\( f(3) \)[/tex] is [tex]\( 19 \)[/tex].
To summarize:
- [tex]\( f(-1) = -1 \)[/tex]
- [tex]\( f(0) = 7 \)[/tex]
- [tex]\( f(3) = 19 \)[/tex]
These are the evaluated values of the piecewise function for the given [tex]\(x\)[/tex] values.
