Problem 78 - College Algebra with Modeling and Visualization

Question

Use \(g(x)\) to complete the following. $$ g(x)=\left\\{\begin{array}{ll} -2 x-6 & \text { if }-8 \leq x \leq-2 \\ x & \text { if }-2

Step-by-Step Solution

Answer

(a) 10, -2, 2, 5; (b) \(-2

Step 1: Evaluating g at specific points

To find \(g(-8)\), use the piece of \(g(x)\) that applies for \(-8 \leq x \leq -2\), which is \(-2x - 6\). Substituting \(x = -8\) yields \(g(-8) = -2(-8) - 6 = 16 - 6 = 10\).For \(g(-2)\), use the same piece: \(g(-2) = -2(-2) - 6 = 4 - 6 = -2\).For \(g(2)\), use the third piece \(0.5x + 1\) since \(2 \leq x \leq 8\). Thus, \(g(2) = 0.5(2) + 1 = 1 + 1 = 2\).Lastly, for \(g(8)\), use \(0.5x + 1\): \(g(8) = 0.5(8) + 1 = 4 + 1 = 5\).

Step 2: Identifying increasing intervals

The function \(g(x)\) is increasing on intervals where the slope of \(g(x)\) is positive. The slopes for each piece are as follows: - For \(-2x - 6\), the slope is \(-2\), which is negative.- For \(x\), the slope is \(1\), which is positive.- For \(0.5x + 1\), the slope is \(0.5\), which is positive.Thus, \(g(x)\) is increasing on the intervals \(-2 < x < 2\) and \(2 \leq x \leq 8\).

Step 3: Sketching and Checking Continuity

To sketch the graph of \(g(x)\), plot each piece: - The line segment \(-2x - 6\) from \(-8\) to \(-2\).- The line \(x\) from \(-2\) to \(2\).- The line segment \(0.5x + 1\) from \(2\) to \(8\).Checking continuity involves ensuring that there are no jumps or gaps between the segments. Since:- \(g(-2) = -2\) from both \(-2x - 6\) and \(x\)- \(g(2) = 2\) from both \(x\) and \(0.5x + 1\)The function \(g\) is continuous across its domain.

Key Concepts

Evaluating Functions

When working with piecewise functions, the first step is often evaluating the function at specific values. A piecewise function is described by different expressions depending on the input value of the variable. In this case, we have three different expressions, each corresponding to different ranges of the variable \(x\).

• For \(x = -8\), we look at the expression \(-2x - 6\), because \(-8\) falls within the interval \(-8 \leq x \leq -2\). Substituting \(-8\) into the expression, we calculate \(g(-8) = 16 - 6 = 10\).
• For \(x = -2\), the same expression \(-2x - 6\) applies, giving \(g(-2) = -2(-2) - 6 = 4 - 6 = -2\).
• When \(x = 2\), we use \(0.5x + 1\), as the condition \(2 \leq x \le 8\) applies. Thus, \(g(2) = 0.5(2) + 1 = 1 + 1 = 2\).
• Finally, for \(x = 8\), with the same expression \(0.5x + 1\), we find \(g(8) = 0.5(8) + 1 = 4 + 1 = 5\).

Evaluating functions this way ensures that for each specific \(x\), you use the right part of the piecewise definition.

Increasing Intervals

The increasing intervals of a piecewise function are found by looking at the slopes of the various linear segments.

• The first segment \(-2x - 6\) has a slope of \(-2\), which is negative. Therefore, the function is decreasing in the interval \(-8 \leq x \leq -2\).
• The second segment, \(x\), has a slope of \(1\), which is positive. Here, the function is increasing over the interval \(-2 < x < 2\).
• Lastly, the expression \(0.5x + 1\) has a slope of \(0.5\), also positive, making it an increasing function on \(2 \leq x \leq 8\).

By identifying the sign of each slope, we can map out where the function is behaving in an increasing manner.

Graph Sketching

Graph sketching of a piecewise function involves plotting each piece according to its defined range.

• Begin with the segment \(-2x - 6\) over \(-8 \leq x \leq -2\). This will be a straight line with a negative slope.
• Next, draw the function \(x\) from \(-2\) to \(2\). It is a slanted line moving upwards since it is a positive slope line.
• The final segment, \(0.5x + 1\), spans from \(2\) to \(8\), starting at the point already calculated for \(x = 2\), moving upwards less steeply compared to \(x\), but still increasing.

Connect each segment smoothly, ensuring that each meets at the proper calculated points. This visual representation helps see how the function behaves over its entire domain.

Function Continuity

Determining the continuity of a piecewise function on its domain means checking for gaps or jumps between segments.

• At \(x = -2\), evaluate the connection between \(-2x - 6\) and \(x\). Since \(g(-2)\) from both expressions equals \(-2\), there is no jump.
• At \(x = 2\), both expressions \(x\) and \(0.5x + 1\) lead to \(g(2) = 2\). Another smooth transition occurs.

Since there are no discontinuities at the boundaries between segments, the function \(g\) is continuous on its entire specified domain, meaning the graph stays connected without sudden changes or breaks.

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Other exercises in this chapter

Problem 77

Solve each inequality numerically. Write the solution set in set-builder or interval notation, and approximate endpoints to the nearest tenth when appropriate.

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Problem 78

Solve the equation (to the nearest tenth) (a) symbolically, (b) graphically, and (c) numerically. $$ 3(\pi-x)+\sqrt{2}=0 $$

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Problem 78

Solve each inequality numerically. Write the solution set in set-builder or interval notation, and approximate endpoints to the nearest tenth when appropriate.

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Problem 79

Solve the equation (to the nearest tenth) (a) symbolically, (b) graphically, and (c) numerically. $$ x-3=2 x+1 $$

View solution

Key Concepts

Evaluating Functions

When working with piecewise functions, the first step is often evaluating the function at specific values. A piecewise function is described by different expressions depending on the input value of the variable. In this case, we have three different expressions, each corresponding to different ranges of the variable \(x\).

• For \(x = -8\), we look at the expression \(-2x - 6\), because \(-8\) falls within the interval \(-8 \leq x \leq -2\). Substituting \(-8\) into the expression, we calculate \(g(-8) = 16 - 6 = 10\).
• For \(x = -2\), the same expression \(-2x - 6\) applies, giving \(g(-2) = -2(-2) - 6 = 4 - 6 = -2\).
• When \(x = 2\), we use \(0.5x + 1\), as the condition \(2 \leq x \le 8\) applies. Thus, \(g(2) = 0.5(2) + 1 = 1 + 1 = 2\).
• Finally, for \(x = 8\), with the same expression \(0.5x + 1\), we find \(g(8) = 0.5(8) + 1 = 4 + 1 = 5\).

Evaluating functions this way ensures that for each specific \(x\), you use the right part of the piecewise definition.

Increasing Intervals

The increasing intervals of a piecewise function are found by looking at the slopes of the various linear segments.

• The first segment \(-2x - 6\) has a slope of \(-2\), which is negative. Therefore, the function is decreasing in the interval \(-8 \leq x \leq -2\).
• The second segment, \(x\), has a slope of \(1\), which is positive. Here, the function is increasing over the interval \(-2 < x < 2\).
• Lastly, the expression \(0.5x + 1\) has a slope of \(0.5\), also positive, making it an increasing function on \(2 \leq x \leq 8\).

By identifying the sign of each slope, we can map out where the function is behaving in an increasing manner.

Graph Sketching

Graph sketching of a piecewise function involves plotting each piece according to its defined range.

• Begin with the segment \(-2x - 6\) over \(-8 \leq x \leq -2\). This will be a straight line with a negative slope.
• Next, draw the function \(x\) from \(-2\) to \(2\). It is a slanted line moving upwards since it is a positive slope line.
• The final segment, \(0.5x + 1\), spans from \(2\) to \(8\), starting at the point already calculated for \(x = 2\), moving upwards less steeply compared to \(x\), but still increasing.

Connect each segment smoothly, ensuring that each meets at the proper calculated points. This visual representation helps see how the function behaves over its entire domain.

Function Continuity

Determining the continuity of a piecewise function on its domain means checking for gaps or jumps between segments.

• At \(x = -2\), evaluate the connection between \(-2x - 6\) and \(x\). Since \(g(-2)\) from both expressions equals \(-2\), there is no jump.
• At \(x = 2\), both expressions \(x\) and \(0.5x + 1\) lead to \(g(2) = 2\). Another smooth transition occurs.

Since there are no discontinuities at the boundaries between segments, the function \(g\) is continuous on its entire specified domain, meaning the graph stays connected without sudden changes or breaks.