P.3 Operations on Functions; Graphing Techniques

1 Form the Sum, Difference, Product, and Quotient of Two Functions

Functions, like numbers, can be added, subtracted, multiplied, and divided. For example, the polynomial function \(F( x) =x^{2}+4x\) is the sum of the two functions \(f( x) =x^{2}\) and \(g( x) =4x.\) The rational function \(R( x) =\dfrac{x^{2}}{x^{3}-1}\), \(x\neq 1\), is the quotient of the two functions \(f( x) =x^{2}\) and \(g( x) =x^{3}-1\).

2 Form a Composite Function

Suppose an oil tanker is leaking, and your job requires you to find the area of the circular oil spill surrounding the tanker. You determine that the radius of the spill is increasing at a rate of \(3\) meters per minute. That is, the radius \(r\) of the spill is a function of the time \(t\) in minutes since the leak began, and can be written as \(r( t) =3t\).

For example, after \(20\) minutes the radius of the spill is \(r( 20) =3\cdot 20=60\) meters. Recall that the area \(A\) of a circle is a function of its radius \(r\); that is, \(A( r) =\pi r^{2}\). So, the area of the oil spill after \(20\) minutes is \(A( 60) =\pi ( 60^{2}) =3600\pi \) square meters. Notice that the argument \(r\) of the function \(A\) is itself a function, and that the area \(A\) of the oil spill is found at any time \(t\) by evaluating the function \(A=A(r( t)) \).

Functions such as \(A=A(r( t)) \) are called composite functions.

Another example of a composite function is \(y=( 2x+3) ^{2}\). If \( y=f( u) =u^{2}\) and \(u=g( x) =2x+3\), then by substituting \(g( x) =2x+3\) for \(u\), we obtain the original function: \[ \begin{array}{l} y = f(u) = f(g(x)) = (2x + 3)^2 \\ {\color{#0066A7}{\,\,\,\,\,\,\,\,\,\,\,\,u\, = \mathop g\limits^ \uparrow (x)\,\,\,g(x)\mathop = \limits^ \uparrow 2x + 3}} \\ \end{array} \]

This substitution process is called composition.

In general, suppose that \(f\) and \(g\) are two functions and that \(x\) is a number in the domain of \(g\). By evaluating \(g\) at \(x\), we obtain \(g( x)\). Now, if \(g( x) \) is in the domain of the function \(f\), we can evaluate \(f\) at \(g( x) ,\) obtaining \(f( g( x) )\). The correspondence from \(x\) to \(f( g( x) )\) is called composition.

Figure 40 illustrates the definition. Only those numbers \(x\) in the domain of \(g\) for which \(g( x) \) is in the domain of \(f\) are in the domain of \(f\circ g\). As a result, the domain of \(f\circ g\) is a subset of the domain of \(g\), and the range of \(f\circ g\) is a subset of the range of \( f \).

In general, the composition of two functions \(f\) and \(g\) is not commutative. That is, \(f\circ g\) almost never equals \(g\circ f\). For example, in Example 3, \[ ( g\circ f) ( x) =g( f( x) ) = \dfrac{4}{f( x) -1}=\dfrac{4}{\dfrac{1}{x+2}-1}=\dfrac{4( x+2) }{1-( x+2) }=-\dfrac{4( x+2) }{x+1} \]

Functions \(f\) and \(g\) for which \(f\circ g\) \(=g\circ f\) will be discussed in the next section.

Some techniques in calculus require us to "decompose" a composite function. For example, the function \(H( x) =\sqrt{x+1}\) is the composition \( f\circ g\) of the functions \(f( x) =\sqrt{x}\) and \(g( x) =x+1\).

Although the functions \(f\) and \(g\) chosen in Example 4 are not unique, there is usually a "natural" selection for \(f\) and \(g\) that first comes to mind. When decomposing a composite function, the "natural" selection for \(g\) is often an expression inside parentheses, in a denominator, or under a radical.

3 Transform the Graph of a Function with Vertical and Horizontal Shifts

At times we need to graph a function that is very similar to a function with a known graph. Often techniques, called transformations, can be used to draw the new graph.

First we consider translations. Translations shift the graph from one position to another without changing its shape, size, or direction.

For example, let \(f\) be a function with a known graph, say, \(f( x) =x^{2}\). If \(k\) is a positive number, then adding \(k\) to \(f\) adds \(k\) to each \(y\)-coordinate, causing the graph of \(f\) to shift vertically up \(k\) units. On the other hand, subtracting \(k\) from \(f\) subtracts \(k\) from each \(y\)-coordinate, causing the graph of \(f\) to shift vertically down \(k\) units. See Figure 41 on page 28.

So, adding (or subtracting) a positive constant to a function shifts the graph of the original function vertically up (or down). Now we investigate how to shift the graph of a function right or left.

Again, let \(f\) be a function with a known graph, say, \(f( x) =x^{2} \), and let \(h\) be a positive number. To shift the graph to the right \( h\) units, subtract \(h\) from the argument of \(f.\) In other words, replace the argument \(x\) of a function \(f\) by \(x-h\), \(h>0.\) The graph of the new function \(y=f( x-h) \) is the graph of \(f\) shifted horizontally right \(h\) units. On the other hand, if we replace the argument \(x\) of a function \(f\) by \(x+h,\) \(h>0\), the graph of the new function \( y=f( x+h) \) is the graph of \(f\) shifted horizontally left \(h\) units. See Figure 42 on page 28.

The graph of a function \(f\) can be moved anywhere in the plane by combining vertical and horizontal shifts.

Notice the points plotted in Figure 43. Using key points can be helpful in keeping track of the transformation that has taken place.

4 Transform the Graph of a Function with Compressions and Stretches

When a function \(f\) is multiplied by a positive number \(a\), the graph of the new function \(y=af( x) \) is obtained by multiplying each \(y\) -coordinate on the graph of \(f\) by \(a\). The new graph is a vertically compressed version of the graph of \(f\) if \( 0<a<1\), and is a vertically stretched version of the graph of \(f\) if \(a>1.\) Compressions and stretches change the proportions of a graph.

For example, the graph of \(f( x) = x^{2}\) is shown in Figure 44(a). Multiplying \(f\) by \(a=\dfrac{1}{2}\) produces a new function \(y=\dfrac{ 1}{2}f( x) =\dfrac{1}{2}x^{2}\), which vertically compresses the graph of \(f\) by a factor of \(\dfrac{1}{2}\), as shown in Figure 44(b). On the other hand, if \(a=3,\) then multiplying \(f\) by \(3\) produces a new function \( y=3~f( x) =3x^{2}\), and the graph of \(f\) is vertically stretched by a factor of 3, as shown in Figure 44(c).

If the argument \(x\) of a function \(f\) is multiplied by a positive number \( a \), the graph of the new function \(y=f( ax) \) is a horizontal compression of the graph of \(f\) when \(a>1\), and a horizontal stretch of the graph of \(f\) when \(0<a<1\).

For example, the graph of \(y=f( 2x) =( 2x) ^{2}=4x^{2}\) is a horizontal compression of the graph of \(f( x) =x^{2}.\) See Figure 45(a) and 45(b). On the other hand, if \(a=\dfrac{1}{3}\), then the graph of \(y=\left( \dfrac{1}{3}x\right) ^{\!\!2}=\dfrac{1}{9}x^{2}\) is a horizontal stretch of the graph of \(f( x) =x^{2}.\) See Figure 45(c).

5 Transform the Graph of a Function by Reflecting It about the x-axis or the y-axis

The third type of transformation, reflection about the \(x\)- or \(y\)-axis, changes the orientation of the graph of the function \(f\) but keeps the shape and the size of the graph intact.

When a function \(f\) is multiplied by \(-1,\) the graph of the new function \( y=-f( x) \) is the reflection about the \(x\)-axis of the graph of \(f\). For example, if \(f( x) =\sqrt{x}\), then the graph of the new function \(y=-f( x) =-\sqrt{x}\) is the reflection of the graph of \(f\) about the \(x\)-axis. See Figures 46(a) and 46(b).

If the argument \(x\) of a function \(f\) is multiplied by \(-1\), then the graph of the new function \(y=f( -x) \) is the reflection about the \(y\)-axis of the graph of \(f\). For example, if \(f( x) =\sqrt{x}\), then the graph of the new function \(y=f( -x) \) \(=\sqrt{-x}\) is the reflection of the graph of \(f\) about the \(y\)-axis. See Figures 46(a) and 46(c). Notice in this example that the domain of \(y=\sqrt{-x}\) is all real numbers for which \(-x\geq 0\), or equivalently, \(x\leq 0.\)