Problem 19 - Calculus Early Transcendentals: Pearson New International Edition

Question

Sketch the graphs of \(f(x)=(x-3) / 2\) and \(g(x)=\sqrt{x}\) using the same coordinate axes. Then sketch \(f+g\) by adding \(y\)-coordinates.

Step-by-Step Solution

Answer

Sketch the graphs of \(f(x) = \frac{x-3}{2}\), \(g(x) = \sqrt{x}\), and \(f+g\) on the same axes.

Step 1: Identify the Functions

We need to sketch the functions \(f(x) = \frac{x-3}{2}\) and \(g(x) = \sqrt{x}\). The first function is a linear function, while the second one is a square root (non-linear) function.

Step 2: Plot Points for f(x)

Choose a few values for \(x\) to find the corresponding \(y\)-values for \(f(x)\).- For \(x=0\), \(f(0) = \frac{0-3}{2} = -\frac{3}{2}\).- For \(x=2\), \(f(2) = \frac{2-3}{2} = -\frac{1}{2}\).- For \(x=4\), \(f(4) = \frac{4-3}{2} = \frac{1}{2}\).- Plot these points and draw a straight line, as the function is linear.

Step 3: Plot Points for g(x)

Choose values for \(x\) (only non-negative values due to the square root) and calculate corresponding \(y\)-values for \(g(x)\).- For \(x=0\), \(g(0) = \sqrt{0} = 0\).- For \(x=1\), \(g(1) = \sqrt{1} = 1\).- For \(x=4\), \(g(4) = \sqrt{4} = 2\).- Plot these points and draw a smooth curve, as the function is non-linear.

Step 4: Plot Points for f+g

Add the y-coordinates from the same \(x\) values in both functions to find and plot points for \(f+g\).- For \(x=0\), \((f+g)(0) = -\frac{3}{2} + 0 = -\frac{3}{2}\).- For \(x=1\), \((f+g)(1) = -1 + 1 = 0\).- For \(x=4\), \((f+g)(4) = \frac{1}{2} + 2 = \frac{5}{2}\).- Plot these points and draw a smooth curve representing \(f+g\).

Step 5: Draw the Combined Graphs

On the same set of coordinate axes, draw the line graph for \(f(x)\), the curve for \(g(x)\), and the resulting curve for \(f+g(x)\). This will visually show how the functions behave individually and when added together.

Key Concepts

Linear Functions

A linear function is one of the simplest types of functions in mathematics. It's called 'linear' because its graph forms a straight line in a coordinate plane. The general form of a linear function is given by \(f(x) = mx + c\), where \(m\) represents the slope and \(c\) is the y-intercept.
In linear functions, the change in \(y\) is proportional to the change in \(x\). This means if you graph any linear function, the slope \(m\) will show how steep or flat the line is.
**For example:** In the function \(f(x) = \frac{x-3}{2}\), it can be rewritten as \(f(x) = \frac{1}{2}x - \frac{3}{2}\), where the slope \(m\) is \(\frac{1}{2}\), showing that for every 2 units the graph runs horizontally, it rises by 1 unit vertically. The y-intercept is \(-\frac{3}{2}\), which is the point where the line crosses the \(y\)-axis.
Linear functions have consistent rates of change, making them very predictable and useful for solving real-world problems.

Non-linear Functions

Non-linear functions describe relationships that are not constant in their rate of change, resulting in graphs that are curves instead of straight lines. One common type of non-linear function is a square root function, like \(g(x) = \sqrt{x}\).
Unlike linear functions, non-linear functions can display varying degrees of change depending on the value of \(x\). This means their graphs can take many shapes, including parabolas, circles, or other complex curves.
**With square root functions:** The graph generally starts at the origin \((0,0)\) because you cannot take the square root of negative numbers in real number systems. As \(x\) increases, \(y\) also increases but at a decreasing rate. This results in a gentle upward curve.
In \(g(x) = \sqrt{x}\), as \(x\) went from 0 to 4 in our earlier example, the \(y\)-values go from 0 to 2, forming a curve that starts steep and gradually flattens.

Coordinate Geometry

Coordinate geometry, or analytic geometry, is a field of mathematics that uses a coordinate system to visualize and analyze geometrical shapes like lines and curves. The most common coordinate system is the Cartesian coordinate system, where each point is defined by a pair \((x, y)\).
This system allows for graphing equations to show how \(x\) and \(y\) are related visually. To graph a function, points are plotted by substituting values for \(x\) and finding the corresponding \(y\) value.
For instance, in graphing functions like \(f(x) = \frac{x-3}{2}\) and \(g(x) = \sqrt{x}\), points noted in earlier steps were connected to display the overall shape of each function. Linear functions make straight lines while non-linear ones like the square root function create curves.
Using this concept, we can predict behavior, find intersections (where graphs cross), and understand relationships between different functions.

Function Addition

Function addition involves combining two functions to create a new function through the addition of their outputs at each value of \(x\). When adding the two functions, \(f(x)\) and \(g(x)\), the resulting function, \((f+g)(x)\), is defined as \(f(x) + g(x)\) for each \(x\).
This concept is visualized on a graph as the addition of their corresponding \(y\)-coordinates.
For example, if we have points like \((x, y_1)\) from \(f(x)\) and \((x, y_2)\) from \(g(x)\), then for \((f+g)(x)\) at the same \(x\), the point becomes \((x, y_1 + y_2)\).
In our exercise, after plotting points for \(f(x)\) and \(g(x)\), such points were used to calculate and plot points for \(f+g\). For instance, at \(x = 4\), the calculation \(\frac{1}{2} + 2 = \frac{5}{2}\) shows how their outputs combine to create the new graph of \(f+g(x)\). This resulting curve provides a visualization of how two functions interact when combined.