Math algebra

MAT-121: COLLEGE ALGEBRA

Written Assignment 3

2 points each except for 5, 6, 9, 15, 16, which are 4 points each as indicated.

SECTION 3.1

Algebraic

For the following exercise, determine whether the relationship represents y as a function of x. If the relationship represents a function then write the relationship as a function of x using f as the function.

1. x+y2=5


2. Consider the relationship 7n-5m=4.
   
   
   
   1. Write the relationship as a function n = k(m).
   
   
   2. Evaluate k(5).
   
   
   3. Solve for k(m) = 7.
   
   
   
   

Graphical

1. Given the following graph
   
   
   
   1. Evaluate f(4)
   
   
   2. Solve for f(x) = 4
   
   
   
   

Numeric

For the following exercise, determine whether the relationship represents a function.

1. {(0, 5), (-5, 8), (0, -8)}

For the following exercise, use the function f represented in table below. (4 points)

x

-18

-12

-6

0

6

12

18

f(x)

24

17

10

3

-4

-11

-18

1. Answer the following:
   
   
   
   1. Evaluate f(-6).
   
   
   2. Solve f(x) = -11
   
   
   3. Evaluate f(12)
   
   
   4. Solve f(x) = -18
   
   
   
   

For the following exercise, evaluate the expressions, given functions f, g, and h:

f(x)=4x+2; g(x)=7-6x; h(x)=7x2-3x+6

1. f(-1)g(1)h(0) (4 points)

Real-world applications

1. The number of cubic yards of compost, C, needed to cover a garden with an area of A square feet is given by C = h(A).
   
   
   
   1. A garden with an area of 5,000 ft2 requires 25 yd3 of compost. Express this information in terms of the function h.
   
   
   2. Explain the meaning of the statement h(2500) = 12.5.
   
   
   
   

SECTION 3.2

Algebraic

For the following exercise, find the domain and range of each function and state it using interval notation.

1. f(x)=9-2x5x+13

Numeric

For the following exercise, given each function f, evaluate f (3), f (-2), f (1), and f (0). (4 points)

Real-World Applications 

1. The height, h, of a projectile is a function of the time, t, it is in the air. The height in meters for t seconds is given by the function h(t)= -9.8t2+19.6t. What is the domain of the function? What does the domain mean in the context of the problem?  

SECTION 3.3

Algebraic

For the following exercise, find the average rate of change of each function on the interval specified in simplest form.

1. k(x)=23x+1on [2, 2+h]

Graphical

For the following exercise, use the graph of each function to estimate the intervals on which the function is increasing or decreasing.

For the following exercise, find the average rate of change of each function on the interval specified.

1. g(x)=3x2-23x3 on [1, 3] 

Real-World Applications

1. Near the surface of the moon, the distance that an object falls is a function of time. It is given by d(t)=1.6t2, where t is in seconds and d(t) is in meters. If an object is dropped from a certain height, find the average velocity of the object from t = 2 to t = 5.

SECTION 3.4

Algebraic

For the following exercise, determine the domain for each function in interval notation. (4 points)

1. f(x)=2x+5 and g(x)=4x+9, find f-g, f+g, fg, and fg

For the following exercise, use each set of functions to findg(f(h(x))). Simplify your answers. (4 points)

1. f(x)=x3-3,g(x)=6-x, and h(x)=2x

Numeric

For the following exercise, use the function values for f and g shown in table below to evaluate each expression.

x

-8

-6

-4

-2

0

2

4

6

8

f(x)

15

8

9

6

3

0

-3

-6

-8

g(x)

-6

-1

3

7

11

15

-8

23

0

1. fg(4)


2. fg(-8)


3. gf(-6)


4. gf(8)

For the following exercise, use each pair of functions to findf(g(-1))andg(f(-1)).

1. f(x)= -62x+5, g(x)= -x2+7

Real-World Applications

1. The radius r, in inches, of a spherical balloon is related to the volume, V, by the following:

r=33V4

Air is pumped into the balloon, so the volume after t seconds is given byV(t)=15+18t. Find the composite function r(V(t)) and use it to answer the following questions. Round the results to the nearest hundredth of a second.

1. At what exact time is the radius 10 inches?  


2. At what exact time is the radius 20 inches?  


3. At what exact time is the radius 30 inches? 

SECTION 3.5

Algebraic

For the following exercise, write a formula for the function obtained when the graph is shifted as described.

1. g(x)=x is shifted up 3 units and to the left 1 unit.

For the following exercise, describe how the graph of the function is a transformation of the graph of the original function f.

1. y=f(x-5)-9

For the following exercise, determine the interval(s) on which the function is increasing and decreasing.

1. h(x)=6(x-8)2+7

Graphical

For the following exercise, sketch a graph of the function as a transformation of the graph of one of the toolkit (basic) functions. State the transformation in words.

1. f(x)=x-32+2

Numeric

For the following exercise, determine whether the function is odd, even, or neither. 

1. f(x)=(x+4)2

For the following exercise, write a formula for the function f that results when the graph of a given toolkit function is transformed as described.

1. The graph of f(x)=x is reflected over the x-axis, shifted vertically up 1 unit, horizontally compressed by a factor of 37.

SECTION 3.6

Algebraic

1. Describe the situation in which the distance that point x is from 18 is at least 13 units. Express this using absolute value notation.

For the following exercise, find the x- and y-intercepts of the graph of the function.

1. f(x)=x+9-11

Real-World Applications

1. A machinist must produce a circular metal brace that is within 0.001 inches of the correct diameter of 6.5 inches. Using x as the diameter of the circular metal brace, write this statement using absolute value notation.

SECTION 3.7

Algebraic

For the following exercise, findf-1(x) for the function.

1. f(x)= -2x-13

For the following exercise, find a domain on which the function f is one-to-one and non-decreasing. Write the domain in interval notation. Then find the inverse of f restricted to that domain.

1. f(x)=x+112-14

Graphical

For the following exercises, use a graphing utility to determine whether each function is one-to-one.

1. f(x)=x-32+6

Numeric

For the following exercise, evaluate or solve, assuming that the function f is one-to-one.

1. If f(23)=32, find f-132.

Real-World Applications

1. The area A of a circle is a function of its radius given by A(r)=r2. Express the radius of a circle as a function of its area. Call this function r(A). Find r(64)and interpret its meaning.

SECTION 4.1

Algebraic

For the following exercise, determine whether the equation of the curve can be written as a linear function.

1. x+6=4y-2

For the following exercise, find the slope of the line that passes through the two given points.

1. (-3, 5) and (6, -11) 

For the following exercise, find a linear equation satisfying the conditions, if possible.

1. x-intercept at (2, 0) and y-intercept at (0, 3)

For the following exercise, determine whether the lines given by the equations below are parallel, perpendicular, or neither.

1. 3x+8y=12 and 15-8y=3x

For the following exercise, find the x- and y-intercepts of the equation.

1. 3x5+79y=23

For the following exercise, use the description of each pair of lines given below to find the slopes of Line 1 and Line 2. Is each pair of lines parallel, perpendicular, or neither?

1. Line 1: Passes through (2, 3) and (4, −1)

Line 2: Passes through (6, 3) and (12, 6) 

For the following exercise, write an equation for the line described.

1. Write an equation for a line perpendicular to p(t)= -11t+14 and passing through the point (−3, −1).

Graphical

For the following exercise, find the slope of the lines graphed.

For the following exercise, sketch a line with the given features.

1. A y-intercept of (0, -7) and slope of 35

For the following exercise, sketch the graph of the equation.

1. g(x)=2x3-12

Real-World Applications

1. At noon, a waitress notices that she has $12 in her tip jar. If she makes an average of $2.25 from each customer, how much will she have in her tip jar if she serves n more customers during her shift?

SECTION 4.2

Algebraic

For the following exercise, consider this scenario:

1. The weight of a newborn is 7.25 pounds. The baby gains38of a pound a month for its first year.
   
   
   
   1. Find the linear function that models the baby’s weight, W, as a function of the age of the baby, in months, t.
   
   
   2. When did the baby weigh 9.875 pounds?
   
   
   3. What is the baby’s weight after 10.5 months?