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Section 3.16
Operations on Functions


The phrase “operate on a function” f means to define a new function that does something with every value of f, with every value of f’s independent variable, or both.

Sum: The sum of two functions must be a function. Suppose you have two functions f and g. A function h, defined as $h(x) = f(x) + g(x)$, is called the sum of f and g. Every value of h is a sum of values of f and g. Take note: A difference of two functions is also a sum.

A sum of two functions is defined only for values where each function is defined. For example, let f be defined as f(x) = 2x,  let g be defined as $g(x) = 1/(x-2)$, and let h be defined as $h(x)=f(x) + g(x)$. Then h(2) is undefined because g(2) is undefined.


Product: The product of two functions must be a function. Suppose you have two functions f and g. A function p, defined as $p(x) = f(x)g(x)$, is called the product of f and g. Every value of p is a product of values of f and g.

A product of two functions is defined only for values where each function is defined. For example, let f be defined as f(x) = 2x,  let g be defined as g(x) = 1/(x-2), and let h be defined as $h(x)=f(x)g(x)$. Then h(2) is undefined because g(2) is undefined.


Quotient: The quotient of two functions is a function. Suppose you have two functions f and g. A function q, defined as $q(x) = \frac{f(x)}{g(x)}$, is called the quotient of f and g. Every value of q is a quotient of values of f and g. If f and g are polynomial functions, then q is called a rational function.

A quotient of two functions f and g is defined only for values where each function is defined and for values of x such that $g(x) ≠ 0$. For example, let f be defined as f(x) = 2x,  let g be defined as g(x) = (x-2), and let h be defined as $h(x)=f(x)/g(x)$. Then h(2) is undefined because g(2) = 0.


Power: A power of a function is a function. Suppose you have two functions f and g. A function s, defined as $s(x) = f(x)g(x)$, is called the power of f and g, or f to the power of g. Every value of s is a power of values of f and g. If f is a constant function, such as $f(x) = 2$ for all values of x, then s is called an exponential function.

A power function is defined only for values where each function is defined and for values of x such that $f(x) > 0$. For example, let f be defined as f(x) = 2x,  let g be defined as g(x) = 1/(x-2), and let h be defined as $h(x)=f(x)^{g(x)}$. Then h(2) is undefined because g(2) is undefined.


Composite: Suppose you have two functions f and g. A function c, defined as $c(x) = g(f(x))$ is called the composite of f and g, or the composition of f and g. Every value of c is the value of g evaluated at a value of f. You have seen many examples of composite functions. When we defined c as $c(x) = \sin(3x + 2)$, we defined a composite function. Think of g as g(x) = sin(x) and think of $3x + 2$ as $f(x) = 3x + 2$. Then $c(x) =g(f(x))$. Any function with a complicated argument is a composite function.

The composite function g(f(x)) is defined only for values of f that are in the domain of g’s independent variable.

The case of a composite function deserves special clarification. In the description of the composite function c, we said “Every value of c is the value of g evaluated at a value of f.” The animation in Figure 3.16.1 is designed to clarify this statement. The animations illustrates one way to think of evaluating $g(f(x))$ at $x = a$. It shows a being assigned as a generic value of x, then evaluating f at $x = a$ to get $f(a)$. Then the value of $f(a)$ is assigned as the value of u, g’s independent variable, getting the value $g(f(a))$ as the value of g at $u = f(a)$. Finally, a, as a value of x, is paired with the value $g(f(a))$.



Figure 3.16.1. Evaluating the composite function c(x) = g(f(x) at x = a.

Exercise Set 3.16

1. How can we think of “2” in the function j, defined as $j(x) = h(x) + 2$, so that j is a sum of two functions instead of a sum of a function and a number?

2. How can we think of e in the function j, defined as $j(x) = e^{h(x)}$, so that j is a power function instead of a number to a power?

3. How can we think of “1” and “x” in $f(x) = \frac{1}{x}$ so that f is a quotient function?

4. The definitions of sum, product, etc. each contain a clause about when it is defined with respect to functions’ independent variables. Construct an example (they can be hand-drawn graphs) for each of sum, product, etc. that illustrates why the given restrictions on independent variables’ values cannot be violated.

5. Use GC to make three examples of each type of function: sum, product, quotient, and power. Make a different file for each type.

6. Here are GC graphs of two functions, f and g. Their domains are restricted to their respective visible intervals. Sketch graphs of $f + g$, $f \cdot g$, $\frac{f}{g}$, $g^f$, and $g(f)$. After sketching all five, check your answers with this file.

             

7. Enter these lines in GC. Why are there gaps in the graph generated by the statement $y’ = g(f(x’))$?

$f(x)=0.4(x-2)(x-1)(x+1)(x+2)$
$g(x)=\sqrt{x}$
$y=f(x)$
$y=g(x)$
$y'=g(f(x'))$

8. Enter the functions f, g, and h in GC. Why does the graph of $y = f(x) + g(x) + h(x)$ appear where it does?

$f(x)=2$ if $|x-1|>2$
$g(x)=1$ if $|x-1|<2$
$h(x)=1.5$ if $|x-2|<2$

9. Enter following lines into GC.

$j(x)=x^2-nx$
$k(x)=x^3-nx$
$y=j(x)$
$y'=k(x')$

Click “n” in the n-slider. Change its lowest value to -2 and its highest value to 2. Click on n’s play button.

a) Explain why the graph of $y = j(x)$ changes the way that it does.

b) Explain why the graph of $y’ = k(x’)$ changes the way that it does.

c) Enter the lines $y = -nx$ and $y’ = -nx’$ into GC and restart the n-slider.

d) Now give one explanation that works for why both graphs change the way they do.