6. Given that $g(x)=\left\{\begin{array}{cl}x+4 & ext { if } x \leq-2 \ 2 & ext { if }-2

Question

6. Given that $g(x)=\left\{\begin{array}{cl}x+4 & 	ext { if } x \leq-2 \ 2 & 	ext { if }-2<x<2 \ 4-x & 	ext { if } x \geq 2\end{array}ight.$,
(a). Find $g(2)$
(b) Find $g(0)$
(c). Find $g(-4)$
(d). Find $g(3)$
(e). Sketch the graph of $g(x)$.

Accepted Answer

∻Solution by Steps∻ ### (a) Find $g(2)$ ‖ step 1 ⋮ Identify the piece of the function $g(x)$ that applies when $x = 2$. According to the definition, for $x \geq 2$, $g(x) = 4 - x$. ‖ ‖ step 2 ⋮ Substitute $x = 2$ into the function: $g(2) = 4 - 2$. ‖ ‖ step 3 ⋮ Simplify the expression: $g(2) = 2$. ‖ ∻Answer∻ ⚹ $g(2) = 2$ ⚹ ∻Key Concept∻ ⚹Piecewise function evaluation⚹ ∻Explanation∻ ⚹To find $g(2)$, we use the piece of the function defined for $x \geq 2$.⚹ ### (b) Find $g(0)$ ‖ step 1 ⋮ Identify the piece of the function $g(x)$ that applies when $x = 0$. According to the definition, for $-2 < x < 2$, $g(x) = 2$. ‖ ‖ step 2 ⋮ Since $0$ falls within the interval $-2 < x < 2$, we have $g(0) = 2$. ‖ ∻Answer∻ ⚹ $g(0) = 2$ ⚹ ∻Key Concept∻ ⚹Piecewise function evaluation⚹ ∻Explanation∻ ⚹To find $g(0)$, we use the piece of the function defined for $-2 < x < 2$.⚹ ### (c) Find $g(-4)$ ‖ step 1 ⋮ Identify the piece of the function $g(x)$ that applies when $x = -4$. According to the definition, for $x \leq -2$, $g(x) = x + 4$. ‖ ‖ step 2 ⋮ Substitute $x = -4$ into the function: $g(-4) = -4 + 4$. ‖ ‖ step 3 ⋮ Simplify the expression: $g(-4) = 0$. ‖ ∻Answer∻ ⚹ $g(-4) = 0$ ⚹ ∻Key Concept∻ ⚹Piecewise function evaluation⚹ ∻Explanation∻ ⚹To find $g(-4)$, we use the piece of the function defined for $x \leq -2$.⚹ ### (d) Find $g(3)$ ‖ step 1 ⋮ Identify the piece of the function $g(x)$ that applies when $x = 3$. According to the definition, for $x \geq 2$, $g(x) = 4 - x$. ‖ ‖ step 2 ⋮ Substitute $x = 3$ into the function: $g(3) = 4 - 3$. ‖ ‖ step 3 ⋮ Simplify the expression: $g(3) = 1$. ‖ ∻Answer∻ ⚹ $g(3) = 1$ ⚹ ∻Key Concept∻ ⚹Piecewise function evaluation⚹ ∻Explanation∻ ⚹To find $g(3)$, we use the piece of the function defined for $x \geq 2$.⚹ ### (e) Sketch the graph of $g(x)$ ‖ step 1 ⋮ Identify the different pieces of the function $g(x)$: - For $x \leq -2$, $g(x) = x + 4$. - For $-2 < x < 2$, $g(x) = 2$. - For $x \geq 2$, $g(x) = 4 - x$. ‖ ‖ step 2 ⋮ Plot the piece $g(x) = x + 4$ for $x \leq -2$. This is a line with a slope of 1 and y-intercept at 4. ‖ ‖ step 3 ⋮ Plot the piece $g(x) = 2$ for $-2 < x < 2$. This is a horizontal line at $y = 2$. ‖ ‖ step 4 ⋮ Plot the piece $g(x) = 4 - x$ for $x \geq 2$. This is a line with a slope of -1 and y-intercept at 4. ‖ ‖ step 5 ⋮ Combine all the pieces to form the complete graph of $g(x)$. ‖ ∻Answer∻ ⚹ Sketch the graph as described in the steps. ⚹ ∻Key Concept∻ ⚹Piecewise function graphing⚹ ∻Explanation∻ ⚹To sketch the graph of $g(x)$, plot each piece of the function over its respective interval and combine them.⚹◊What is the value of $g(0)$ in the function $g(x)$ provided in the image?⍭ Generate me a similar question◊

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