Do These Quickly
Q1. What transformation of f is represented by g(x) = 3f(x)?
Q2. What transformation of f is represented by h(x) = 5 + f(x)?
Q3. If g is a horizontal translation of f by –4 spaces, then g(x) = .
25 DEFINITION AND PROPERTIES: Composite Function
The composite function f g (pronounced “ f composition g”) is the function (f g)(x) = f(g(x))
The domain of f g is the set of all values of x in the domain of g for which g(x) is in the domain of f.
Function g is called the inside function and is performed on x first.
Function f is called the outside function and is performed on g(x), the value assigned to x by g.
Q4. If p is a vertical dilation of f by a factor of 0.2, then p(x) = .
Q5. Why is y = 3x5not an exponential function, even though it has an exponent?
Q6. Write the general equation of a quadratic function.
Q7. For what value of x will the graph of have a discontinuity?
Q8. Simplify: x3x5
Q9. Find 40% of 300.
Q10. Which of these is a horizontal dilation by a factor of 2?
A. g(x) = 2f(x) C. g(x) = f(0.5x) B. g(x) = 0.5f(x) D. g(x) = f(2x) 1. Flashlight Problem: You shine a flashlight,
making a circular spot of light with radius 5 cm on a wall. Suppose that, as you back away from the wall, the radius of the spot of light increases at a rate of 7 cm/sec.
b. Find the radius of the spot of light when t = 4 and when t = 10 sec. Use the answers to find the area of the wall illuminated at these two times.
c. The area, A(t), illuminated by the light is a function of the radius, and the radius is a function of t. Write an equation for the composite function A(R(t)). Simplify.
d. At what time t will the area illuminated be 50,000 cm2?
2. Bacterial Culture Problem: When you grow a culture of bacteria in a petri dish, the area of the culture is a function of the number of bacteria present. Suppose that the area of the culture, A(t), measured in square millimeters, is given by this function of time, t, measured in hours:
A(t) = 9(1.1t)
a. Find the area at times t = 0, t = 5, and t = 10.
Is the area changing at an increasing rate or
at a decreasing rate? How do the values of A(0), A(5), and A(10) allow you to answer this question?
b. Assume that the bacteria culture is circular.
Use the results of part a to find the radius of the culture at these three times. Is the radius changing at an increasing rate or at a decreasing rate?
c. The radius of the culture is a function of the area. Write an equation for the composite function R(A(t)).
d. The radius of the petri dish is 30 mm. The culture is centered in the dish and grows uniformly in all directions. What restriction does this fact place on the domain of t for the function R A?
3. Two Linear Functions Problem 1: Let f and g be defined by
f(x) = 9 – x 4 x 8 g(x) = x + 2 1 x 5
Figure 1-4f
a. The graphs of f and g are shown in Figure 1-4f. Plot these graphs on your grapher, making sure they agree with the figure.
a. Write an equation for R(t), the radius of the spot of light t sec after you started backing away.
On the same screen, plot the graphs of y = f(g(x)) and y = g(f(x)). Sketch the graphs, showing the domains of each. You may use a copy of Figure 1-4f.
b. Find f(g(3)). Show on your sketch from part a the two steps by which you can find this value directly from the graphs of f and g.
c. Explain why f(g(6)) is undefined. Explain why f(g(1)) is undefined, even though g(1) is defined.
d. Find an equation for f(g(x)) explicitly in terms of x. Simplify as much as possible.
e. Find the domains of both f g and g f algebraically. Do they agree with your graphs in part a?
f. Find the range of f g algebraically. Does it agree with your graph in part a?
g. Find f(f(5)). Explain why g(g(5)) is undefined.
4. Quadratic and Linear Function Problem: Let f and g be defined by
f(x) = –x2 + 8x –4 1 x 6 g(x) = 5 – x 0 x 7
a. The graphs of f and g are shown in Figure 1-4g. Plot these graphs on your grapher. On the same screen, plot the graph of y = f(g(x)). Sketch the resulting graph, showing the domain and range.
You may use a copy of Figure 1-4g.
b. Show that f(g(3)) is defined, but g(f(3)) is undefined.
c. Find the domain of f g algebraically. Show that it agrees with your graph in part a.
d. Find an equation for f(g(x)) explicitly in terms of x. Simplify as much as possible.
Plot the graph of this equation on the same
screen as your graph in part a, observing the domain in part c. Does the graph coincide with the one you plotted in part a?
5. Square Root and Linear Function Problem: Let f and g be defined by
where the values of x make f(x) a real number g(x) = 2 –x where x is any real number
Figure 1-4h
a. Plot the graphs of f and g. Use a friendly x-window that includes x = 4 as a grid point. terms of x. Simplify as much as possible.
Explain why the domain of f g is x –2.
e. Find an equation for g(f(x)). What is the domain of g f?
6. Square Root and Quadratic Function Problem:
Let f and g be defined by
where the values of x make f(x) a real number g(x) = x2 –4 where x is any real number a. Plot the graphs of f, g, and f g on the same
screen. Use a friendly x-window of about –10 x 10 that includes the integers as grid points. Sketch the result.
b. Find the domain of f g . Explain why the domain of f g has positive and negative numbers, whereas the domain of f has only positive numbers.
c. Show algebraically that f(g(4)) is defined, but f(g(1)) is undefined.
27
Figure 1-4g
d. Show algebraically that g(f(6)) is defined, but g(f(3)) is undefined.
e. Plot the graph of g f. Make sure that the window has negative and positive y-values.
f. The graph of g f appears to be a straight line. By finding an equation for g(f(x)) explicitly in terms of x, determine whether this is true.
7. Square and Square Root Functions: Let f and g be defined by
b. Test your conjecture by finding f(g( 9)) and g(f( 9)). Does your conjecture hold for negative values of x?
c. Plot f(x), g(x), and f(g(x)) on the same screen. Use approximately equal scales on both axes, as shown in Figure 1-4i. Explain why f(g(x)) = x, but only for nonnegative values of x.
d. Deactivate f(g(x)), and plot f(x), g(x), and g(f(x)) on the same screen. Sketch the result.
e. Explain why g(f(x)) = x for nonnegative values of x, but –x (the opposite of x) for negative values of x. What other familiar function has this property?
and g be defined by
g(x) = 1.5x + 3
a. Find f(g(6)), f(g(–15)), g(f(10)), and g(f(–8)).
What do you notice in each case?
c. Show algebraically that f(g(x)) and g(f(x)) both equal x.
d. Functions f and g in this problem are called inverses of each other. Whatever f “does to x,”g “undoes.” Let h(x) = 5x –7. Find an equation for the function that is the inverse of h.
For Problems 9 and 10, find what transformation will transform f (dashed graph) into g (solid graph).
11. Horizontal Translation and Dilation Problem:
Let f, g, and h be defined by
f(x) = x2 for –2 x 2
g(x) = x –3 for all real values of x for all real values of x a. f(g(x)) = f(x –3). What transformation of
function f is equivalent to the composite function f g?
Linear Function and Its Inverse Problem: Let f
Figure 1-4i