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The phrase “operate on a function”

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

2. How can we think of

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

4. The definitions of

5. Use GC to make three examples of each type of function:

6. Here are GC graphs of two functions,

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

$f(x)=0.4(x-2)(x-1)(x+1)(x+2)$

$g(x)=\sqrt{x}$

$y=f(x)$

$y=g(x)$

$y'=g(f(x'))$

$g(x)=\sqrt{x}$

$y=f(x)$

$y=g(x)$

$y'=g(f(x'))$

8. Enter the functions

$f(x)=2$ if $|x-1|>2$

$g(x)=1$ if $|x-1|<2$

$h(x)=1.5$ if $|x-2|<2$

$g(x)=1$ if $|x-1|<2$

$h(x)=1.5$ if $|x-2|<2$

9. Enter following lines into GC.

$j(x)=x^2-nx$

$k(x)=x^3-nx$

$y=j(x)$

$y'=k(x')$

$k(x)=x^3-nx$

$y=j(x)$

$y'=k(x')$

Click “n” in the

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.

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.