QuestionPlot the piecewise function $g(x) = \begin{cases} -6 & \text{if } x < -2 \\ (x+1)^{2} & \text{if } x \geq -2 \end{cases}$ and identify where it increases or decreases.

Studdy Solution

STEP 1

Assumptions1. The function is defined piecewise with two pieces - $g(x) = -6$ for $x < -$ - $g(x) = (x+1)^{}$ for $x \geq -$ . We need to plot the function and find the intervals where the function is increasing and decreasing.

STEP 2

First, let's plot the function. We'll start by plotting the piece $g(x) = -6$ for $x < -2$ . This is a constant function and will be a horizontal line at $y = -6$ for all $x < -2$ .

STEP 3

Next, we'll plot the piece $g(x) = (x+1)^{2}$ for $x \geq -2$ . This is a quadratic function and will be a parabola opening upwards with the vertex at $x = -1$ .

STEP 4

Now that we have the plot, let's find the intervals where the function is increasing and decreasing.

STEP 5

For the first piece $g(x) = -$ for $x < -2$ , the function is constant and hence neither increasing nor decreasing.

STEP 6

For the second piece $g(x) = (x+1)^{2}$ for $x \geq -2$ , the function is a parabola opening upwards. Hence, it is decreasing for $x < -1$ and increasing for $x > -1$ .

STEP 7

However, we need to consider that this piece is only defined for $x \geq -2$ . Hence, the function is decreasing for $-2 \leq x < -1$ and increasing for $x > -1$ .
So, the function is decreasing in the interval $[-2, -1)$ and increasing in the interval $(-1, \infty)$ .

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QuestionPlot the piecewise function $g(x) = \begin{cases} -6 & \text{if } x < -2 \\ (x+1)^{2} & \text{if } x \geq -2 \end{cases}$ and identify where it increases or decreases.

Studdy Solution

STEP 1

Assumptions1. The function is defined piecewise with two pieces - $g(x) = -6$ for $x < -$ - $g(x) = (x+1)^{}$ for $x \geq -$ . We need to plot the function and find the intervals where the function is increasing and decreasing.

STEP 2

First, let's plot the function. We'll start by plotting the piece $g(x) = -6$ for $x < -2$ . This is a constant function and will be a horizontal line at $y = -6$ for all $x < -2$ .

STEP 3

Next, we'll plot the piece $g(x) = (x+1)^{2}$ for $x \geq -2$ . This is a quadratic function and will be a parabola opening upwards with the vertex at $x = -1$ .

STEP 4

Now that we have the plot, let's find the intervals where the function is increasing and decreasing.

STEP 5

For the first piece $g(x) = -$ for $x < -2$ , the function is constant and hence neither increasing nor decreasing.

STEP 6

For the second piece $g(x) = (x+1)^{2}$ for $x \geq -2$ , the function is a parabola opening upwards. Hence, it is decreasing for $x < -1$ and increasing for $x > -1$ .

STEP 7

However, we need to consider that this piece is only defined for $x \geq -2$ . Hence, the function is decreasing for $-2 \leq x < -1$ and increasing for $x > -1$ .
So, the function is decreasing in the interval $[-2, -1)$ and increasing in the interval $(-1, \infty)$ .

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QuestionPlot the piecewise function g(x)={−6if x<−2(x+1)2if x≥−2g(x) = \begin{cases} -6 & \text{if } x < -2 \\ (x+1)^{2} & \text{if } x \geq -2 \end{cases}g(x)={−6(x+1)2​if x<−2if x≥−2​ and identify where it increases or decreases.

Studdy Solution

STEP 1

Assumptions1. The function is defined piecewise with two pieces - g(x)=−6g(x) = -6g(x)=−6 for x<−x < -x<− - g(x)=(x+1)g(x) = (x+1)^{}g(x)=(x+1) for x≥−x \geq -x≥− . We need to plot the function and find the intervals where the function is increasing and decreasing.

STEP 2

First, let's plot the function. We'll start by plotting the piece g(x)=−6g(x) = -6g(x)=−6 for x<−2x < -2x<−2. This is a constant function and will be a horizontal line at y=−6y = -6y=−6 for all x<−2x < -2x<−2.

STEP 3

Next, we'll plot the piece g(x)=(x+1)2g(x) = (x+1)^{2}g(x)=(x+1)2 for x≥−2x \geq -2x≥−2. This is a quadratic function and will be a parabola opening upwards with the vertex at x=−1x = -1x=−1.

STEP 4

Now that we have the plot, let's find the intervals where the function is increasing and decreasing.

STEP 5

For the first piece g(x)=−g(x) = -g(x)=− for x<−2x < -2x<−2, the function is constant and hence neither increasing nor decreasing.

STEP 6

For the second piece g(x)=(x+1)2g(x) = (x+1)^{2}g(x)=(x+1)2 for x≥−2x \geq -2x≥−2, the function is a parabola opening upwards. Hence, it is decreasing for x<−1x < -1x<−1 and increasing for x>−1x > -1x>−1.

STEP 7

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QuestionPlot the piecewise function $g(x) = \begin{cases} -6 & \text{if } x < -2 \\ (x+1)^{2} & \text{if } x \geq -2 \end{cases}$ and identify where it increases or decreases.

Assumptions1. The function is defined piecewise with two pieces - $g(x) = -6$ for $x < -$ - $g(x) = (x+1)^{}$ for $x \geq -$ . We need to plot the function and find the intervals where the function is increasing and decreasing.

First, let's plot the function. We'll start by plotting the piece $g(x) = -6$ for $x < -2$ . This is a constant function and will be a horizontal line at $y = -6$ for all $x < -2$ .

Next, we'll plot the piece $g(x) = (x+1)^{2}$ for $x \geq -2$ . This is a quadratic function and will be a parabola opening upwards with the vertex at $x = -1$ .

For the first piece $g(x) = -$ for $x < -2$ , the function is constant and hence neither increasing nor decreasing.

For the second piece $g(x) = (x+1)^{2}$ for $x \geq -2$ , the function is a parabola opening upwards. Hence, it is decreasing for $x < -1$ and increasing for $x > -1$ .