Precalculus Study Guide
Copyright Maplesoft, a division of Waterloo Maple Inc., 2021
Chapter 5: The Arithmetic of Functions
In this chapter, the arithmetic of functions will be explored. Two functions f⁡x and g⁡x having a common domain can be added, subtracted, multiplied, and divided, to form new functions
Our notation expresses the value of the functions at x, and thereby emphasizes the pointwise definitions used for the arithmetic of functions.
To prescribe one of the functions hk⁡x, we have to prescribe the recipe by which the values of hk⁡x are to be computed. The notation shows, for example, that the value of h1⁡x is simply the sum of the two numbers f⁡x and g⁡x. This is the recipe at each point x in the common domains of f⁡x and g⁡x.
Similarly for the functions h2⁡x and h3⁡x. The only difficulty encountered in defining h4⁡x is at zeros of g⁡x where the fraction f⁡xg⁡x would fail to be defined. Thus, the domains for hk⁡x,k=1,2,3, will be the same as the common domain of f⁡x and g⁡x, but the domain of h4⁡x can be smaller.
The domain of hk⁡x,k=1,...,4, cannot be larger than the common domain of f⁡x and g⁡x. For example, suppose the functions
are added. The rule for the sum would clearly be h⁡x=3, but the domain would not be all real numbers since the common domain for f⁡x and g⁡x was all the reals with the exception of x=0. That would still be the domain of the sum h⁡x.
The following terms in Chapter 5 are linked to the Maple Math Dictionary.
axis of symmetry
Let f⁡x=2⁢x+1 and g⁡x=x2+x−1 be the rules for two functions whose common domain is the set of all real numbers. In Problems 5.1 - 5.4, obtain the indicated arithmetic expression for the function h⁡x, draw its graph, and determine its domain and range. In addition, compute h⁡3 and show that this value can also be obtained from the appropriate combination of the numbers f⁡3 and g⁡3.
Before accessing the Maple version of the solutions to the typical problems stated above, initialize Maple by pressing the button provided on the right.
5.1 - Mathematical Solution
If f⁡x=2⁢x+1, and g⁡x=x2+x−1, with all the reals as the common domain, then the rule for h⁡x=f⁡x+g⁡x is given by
and its graph is given in Figure 5.1.1.
The domain for h⁡x is again the set of all real numbers, while the range is the set of real numbers y for which y>−94. One way to determine the minimum value of y in the range of h⁡x is to find the axis of symmetry for the parabola that is the graph of h⁡x.
Figure 5.1.1 Graph of h⁡x=x2+3⁢x
This axis of symmetry is midway between the x-intercepts, namely, x=0 and x=−3. Hence, the axis of symmetry is y=−32, and the vertex of the parabola is the point ⁡−32,h⁡−32=⁡−32,−94.
Since f⁡3=7 and g⁡3=11, we have h⁡3=18 = 7+11, so h⁡3=f⁡3+g⁡3.
5.1 - Maplet Solution
The arithmetic sum of the functions
as well as its graph, its domain and range, and the function values h⁡3,f⁡3, and g⁡3 are provided by the
Arithmetic of Functions
Clicking this link will launch the tutor with the solution embedded as shown in the thumbnail image in Figure 5.1.2.
Figure 5.1.2 Thumbnail image of the Arithmetic of Functions Tutor
The rule for the sum is
From the graph of h⁡x, we deduce that the domain is the set of all real numbers, and the range is the set of real numbers y greater than or equal to −94. One way to determine the minimum value of y in the range of h⁡x is to find the axis of symmetry for the parabola that is the graph of h⁡x. This axis of symmetry is midway between the x-intercepts, namely, x=0 and x=−3. Hence, the axis of symmetry is y=−32, and the vertex of the parabola is the point ⁡−32,h⁡−32=⁡−32,−94.
We can also see that because f⁡3=7 and g⁡3=11,
h⁡3=18 = 7+11
To launch the Arithmetic of Functions Tutor, click the following link:
Arithmetic of Functions Tutor
5.1 - Interactive Solution
Enter the given data
Control-drag the equation fx=…
Context Panel: Assign Function
Control-drag the equation gx=…