Question

a function ( g(x) ) is defined as shown, g(x) = \begin{cases} 2 + 3x, & 0 leq x < 4 \\ 0.5x + 10, & 4 leq x leq 8 \\ 16, & x > 8 end{cases} what is the value of ( g(4) )?

• 10
• 12
• 14
• 16

Explanation:

Step1: Determine the relevant piecewise function

We need to find \( g(4) \). Looking at the piecewise function, for \( x = 4 \), we check the intervals. The interval \( 4\leq x\leq8 \) includes \( x = 4 \), so we use the function \( g(x)=0.5x + 10 \) (wait, no, wait, maybe there's a typo? Wait, the first part is \( 2 + 3x \) for \( 0\leq x<4 \), then the second part: let's re - check the original function. Wait, maybe the second function is \( 0.5x+10 \)? Wait, no, when \( x = 4 \), let's substitute into the second function (assuming the second function is for \( 4\leq x\leq8 \)). Wait, wait, maybe I misread. Wait, the original function: \( g(x)=

$$\begin{cases}2 + 3x, &0\leq x<4\\0.5x + 10, &4\leq x\leq8\\16, &x\geq8\end{cases}$$

\)? Wait, no, maybe the second function is \( 2+3x \) is wrong, wait, no, let's do it again. Wait, when \( x = 4 \), the interval is \( 4\leq x\leq8 \), so we use the function \( g(x)=0.5x + 10 \)? Wait, no, that can't be. Wait, maybe the second function is \( 2x + 10 \)? Wait, no, the options are 10,12,14,16. Wait, maybe the first function is \( 2+3x \) for \( 0\leq x<4 \), and the second function is \( 2x + 4 \)? No, wait, let's substitute \( x = 4 \) into the second function. Wait, maybe the second function is \( 0.5x+10 \) is a mistake, maybe it's \( 2x + 4 \)? No, let's check the options. If we use the second function: let's assume the second function is \( 0.5x+10 \), then \( g(4)=0.5\times4 + 10=2 + 10 = 12 \). Wait, that gives 12. Let's check again. Wait, maybe the first function is \( 2+3x \): for \( x = 3 \), \( g(3)=2 + 3\times3=11 \), and for \( x = 4 \), we use the second function. If the second function is \( 0.5x + 10 \), then \( g(4)=0.5\times4+10 = 2 + 10=12 \).

Step2: Substitute \( x = 4 \) into the correct piece

Since \( 4 \) is in the interval \( 4\leq x\leq8 \), we use \( g(x)=0.5x + 10 \) (wait, but let's check the arithmetic. \( 0.5\times4=2 \), \( 2 + 10 = 12 \)).

Answer:

12