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Hotmath Practice Problems

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Hotmath Calculus
Hotmath Team
Chapter:Relations and FunctionsSection:Exercises

Problem: 1

Write an equation of the line through point (2, 1) that is perpendicular to 4x – 2y = 3.


Problem: 2

The value of a product in 1998 is $20,400. The value is expected to decrease $2000 per year during the next 5 years. Write a linear equation that gives the dollar value V of the product in terms of the year t. (Let t = 8 represent 1998.)


Problem: 3

Use a graphing utility to graph the parabolas and find their points of intersection. Find an equation of the line through the points of intersection and sketch its graph in the same viewing rectangle.

y = x2

y = 4xx2


Problem: 4

Find an equation of the graph that consists of all the points (x, y) whose distance from the origin is K times (K 1) the distance from (2, 0).


Problem: 5

If f(x) = 2x2 + 3x – 4, find f(0), f(2), f(√ 2), f(1 + √ 2), f(–x), f(x + 1), 2f(x), and f(2x).


Problem: 6

Find the domain and sketch the graph of the function.


Problem: 7

Express the surface area of an open rectangular box with volume 3 m3 and a square base as a function of the length of a side of the base. State the domain of the function.


Problem: 8

Biologists have found that the chirping rate of crickets is related to temperature, and that the relationship is close to linear. If one species of cricket chirps 113 times per minute at 70F and 173 times per minute at 80F, find a linear equation that models the temperature T as a function of the number N of chirps per minute. What does the slope of this line represent?


Problem: 9

Graph the equation

y = x2 – 4x + 3

using transformations of the graph of y = x2.


Problem: 10

If f(x) = 2x2x and g(x) = 3x + 2, find f g, g f, f f, and g g, and their domains.


Problem: 11


g(x) = x2 + 2, and

h(x) = x + 3,

find f g h.


Problem: 12

Use transformations to draw a graph of y = –sin 2x.


Problem: 13

Express the function

as a composition of three functions.


Problem: 14

Show that f and g are inverse functions (a) algebraically and (b) graphically.

f(x) = 5x + 1

g(x) = (x – 1)/5


Problem: 15

Find the inverse function of f over the indicated interval. Use a graphing utility to graph f and f –1 in the same viewing rectangle. Describe the relationship between the graphs.


Problem: 16

Sketch the graph of the function.

y = arcsin (x – 1)


Problem: 17

Solve the equation for x.


Problem: 18

Restrict the domain of f to an infinite interval on which f is one–to–one. Find the inverse of the restricted function and give the domain of the inverse. (Note: there's more than one answer!)

f(x) = (x – 3)2