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# Solutions to Various Algebra Problems

Please solve all odd number problems.

#### Solution Preview

15. {(2, 10), (3, 15), (4, 20)} is a function since there is a unique element on the right of each pair corresponding to each element on the left. The domain (the set of elements on the left side of each pair) is {2, 3, 4} and the range (the set of elements on the right side of each pair) is {10, 15, 20}.

17. {(1, 3), (1, 5), (1, 7), (1, 9)} is not a function since there is more than one element on the right corresponding to the element 1 on the left of each pair. The domain is {1} and the range is {3, 5, 7, 9}.

19. {(-2, 1), (0, 1), (2, 1), (4, 1), (-3, 1)} is a function since there is a unique element on the right of each pair corresponding to each element on the left. The domain is {-3, -2, 0, 2, 4} and the range is {1}.

21. Given g(x) = 3x^2 - 2x + 1

a) g(0) = 3(0^2) - 2(0) + 1 = 1.

b) g(-1) = 3(-1)^2 - 2(-1) + 1
= 3(1) + 2 + 1
= 6.

c) g(3) = 3(3^2) - 2(3) + 1
= 3(9) - 6 + 1
= 27 - 6 + 1
= 22.

d) g(-x) = 3(-x)^2 - 2(-x) + 1
= 3x^2 + 2x + 1.

e) g(1 - t) = 3(1 - t)^2 - 2(1 - t) + 1
= 3(1 - 2t + t^2) - 2 + 2t + 1
= 3 - 6t + 3t^2 - 1 + 2t
= 2 - 4t + 3t^2.

23. Given g(x) = x^3

a) g(2) = 2^3 = 8.

b) g(-2) = (-2)^3 = -8.

c) g(-x) = (-x)^3 = -x^3.

d) g(3y) = (3y)^3 = 27y^3.

e) g(2 + h) = (2 + h)^3
= 2^3 + 3*2^2*h + 3*2*h^2 + h^3
= 8 + 12h + 6h^2 + h^3.

25. Given g(x) = (x - 4)/(x + 3)

a) g(5) = (5 - 4)/(5 + 3) = 1/8.

b) g(4) = (4 - 4)/(4 + 3) = 0/7 = 0.

c) g(-3) = (-3 - 4)/(-3 + 3) = -7/0 is undefined.

d) g(-16.25) = (-16.25 - 4)/(-16.25 + 3)
= -20.25/(-13.25)
= -81/53.

e) g(x + h) = (x + h - 4)/(x + h + 3).

27. Given g(x) = x/sqrt(1 - x^2)

g(0) = 0/sqrt(1 - 0^2) = 0/1 = 0.

g(-1) = -1/sqrt(1 - (-1)^2)
= -1/sqrt(1 - 1)
= -1/0
is undefined.

g(5) = 5/sqrt(1 - 5^2)
= 5/sqrt(-24)
is undefined over the real numbers.

g(1/2) = (1/2) / sqrt(1 - (1/2)^2)
= (1/2) / sqrt(1 - 1/4)
= (1/2) / sqrt(3/4)
= 1/sqrt(3).

Review Exercises:

1. If a < 0, then |a| = -a. This is true, since the absolute value of every negative numbers is positive (the negative of a negative).

3. If a = b, then a + c = b + c. This is true (add c to both sides).

5-9. Of the numbers

-7, 43, -4/9, sqrt(17), 0, ...

#### Solution Summary

The solution solves a host of high school algebra problems.

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