Questions: Evaluate the piecewise function at the given values of the independent variable. f(x) = 3x+5 if x<0 4x+6 if x ≥ 0 (a) f(-2) (b) f(0) (c) f(3) (a) f(-2)= (b) f(0)= (c) f(3)=

Transcript text: Evaluate the piecewise function at the given values of the independent variable. \[ f(x)=\left\{\begin{array}{ll} 3 x+5 & \text { if } x<0 \\ 4 x+6 & \text { if } x \geq 0 \end{array}\right. \] (a) $f(-2)$ (b) $f(0)$ (c) $f(3)$ (a) $f(-2)=$ $\square$ (b) $f(0)=$ $\square$ (c) $f(3)=$ $\square$

Solution

Solution Steps

To evaluate the piecewise function at the given values of the independent variable, we need to determine which part of the function to use based on the value of $ x $. Specifically:

For $ x < 0 $, use $ f(x) = 3x + 5 $.
For $ x \geq 0 $, use $ f(x) = 4x + 6 $.

We will then substitute the given values of $ x $ into the appropriate part of the function to find $ f(-2) $, $ f(0) $, and $ f(3) $.

Step 1: Evaluate $ f(-2) $

Since $ -2 < 0 $, we use the first part of the piecewise function: \[ f(-2) = 3(-2) + 5 = -6 + 5 = -1 \]

Step 2: Evaluate $ f(0) $

Since $ 0 \geq 0 $, we use the second part of the piecewise function: \[ f(0) = 4(0) + 6 = 0 + 6 = 6 \]

Step 3: Evaluate $ f(3) $

Since $ 3 \geq 0 $, we again use the second part of the piecewise function: \[ f(3) = 4(3) + 6 = 12 + 6 = 18 \]

Final Answer

The evaluations yield:

$ f(-2) = -1 $
$ f(0) = 6 $
$ f(3) = 18 $

Thus, the final answers are: \[ \boxed{f(-2) = -1} \] \[ \boxed{f(0) = 6} \] \[ \boxed{f(3) = 18} \]

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