Problem 3 - Calculus: An Integrated Approach to Functions and Their Rates of Change

Question

Let \(f(x)=|x|\) and \(g(x)=x\). Use your knowledge of the graphs of \(f\) and \(g\) to sketch the graph of \(h(x)=f(x)+g(x)\).

Step-by-Step Solution

Answer

The graph of \(h(x)=f(x)+g(x)\) is a straight line of slope 2 for \(x>0\) and the x-axis for \(x<0\).

Step 1: Graph the Function \(f(x)=|x|\)

First, draw the graph of \(f(x)=|x|\). This graph is a v-shape that intersects the origin (0,0) and opens upwards. The slope of the line to the right of the y-axis (for positive x) is 1, and the slope of the line to the left of the y-axis (for negative x) is -1.

Step 2: Graph the Function \(g(x)=x\)

Next, draw the graph of \(g(x)=x\). This graph is a straight line passing through the origin (0,0) with a slope of 1.

Step 3: Combine the Functions to Get \(h(x)=f(x)+g(x)\)

Now, to get the graph of \(h(x)=f(x)+g(x)\), add the y-values of \(f\) and \(g\) for each x. For \(x>0\), both \(f(x)\) and \(g(x)\) are the same so their sum is \(2x\). For \(x<0\), \(f(x)=-x\) and \(g(x)=x\), so their sum is 0. Therefore, the graph of \(h(x)\) is a straight line of slope 2 for \(x>0\) and the x-axis for \(x<0\).

Key Concepts

Absolute Value Function

The absolute value function, represented as \(f(x) = |x|\), is a fascinating type of function because it turns all negative input values into positive output values, while leaving positive values unchanged. Imagine the absolute value function as a mirror that reflects negative values to positive ones. This results in a characteristic V-shaped graph that perfectly straddles the y-axis at the origin \((0, 0)\).

For the absolute value graph:

• The slope is 1 for \(x > 0\), where the graph rises upwards.
• The slope is -1 for \(x < 0\), where it runs downwards towards the left.
• The vertex, or the point of the 'V', is at the origin.

This function is continuous and reflects across the y-axis, making it symmetric. It is a clear and powerful visual aid when understanding the changes in sign of \(x\).

Linear Function

Linear functions, like \(g(x) = x\), are the simplest type of function you can deal with. They produce a straight line when graphed on a coordinate plane. For the function \(g(x) = x\), the line passes through the origin \((0, 0)\) and has a constant slope.

In the case of \(g(x) = x\):

• The slope, or steepness of the line, is 1 which means it rises one unit up for every unit it moves to the right.
• There is no vertical or horizontal shift, as indicated by the fact it passes through \((0, 0)\).

Linear functions are very predictable, showing a direct relationship between input and output, making them foundational in algebra. They are essential in understanding more complex functions by revealing simple trends and changes.

Graph Sketching

Graph sketching is an essential skill for visualizing mathematical relationships between variables. It involves plotting points or equations to see how two functions interact visually. To graph the composition of functions \(h(x) = |x| + x\), you must first understand how to graph each individual function before combining them.

Key steps for graph sketching:

• Graph \(f(x) = |x|\) creating a V-shaped plot.
• Graph \(g(x) = x\), a line through the origin with slope 1.
• For \(h(x) = f(x) + g(x)\), add the y-values of both functions for each x-value.

For \(h(x)\), observe:

• For \(x > 0\), both \(f(x)\) and \(g(x)\) contribute equally, resulting in a line with slope 2, since \(h(x) = 2x\).
• For \(x < 0\), \(f(x) = -x\) exactly cancels out \(g(x) = x\), leading to \(h(x) = 0\), a flat line along the x-axis.

Graph sketching visually demonstrates how functions combine, allowing you to predict behavior easily. It is not only a drawing task but also an interpretation of mathematical operations and interactions.