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Precalculus Problems. (big bonus for...

Precalculus Problems. (big bonus for great solutions)

1. Solve the following equation for x:

log2(x + 7) + log2(x + 8) = 1

Note: Be sure to plug any solutions back into the original equation; if they don't work (e.g., you're forced to take the logarithm of a negative number, which is not possible), discard that solution.

2. The volume V(in cubic meters) of a hot-air balloon is given by V = (4/3) * Pi * R^3.

If the radius r is increasing with time t (in seconds) according to the formula r(t) = 2/3*t^3 , t ≥ 0, find the volume V as a function of the time t.

Solution: V(r(t))=V(2/3*t^3)=4/3*pi(2/3*t^3)^3=4/3*pi(8/27*t^9)=32/81*pi*t^9

3. Find the distance d(P1,P2) between the points P1 and P2.

P1=(1,-3);P2=(2,6)

4. Find the midpoint of the line segment joining the points P1 and P2.

P1=(1,2); P2=(-4,0)

5. Find the equation of a line that is parallel to the line x = 14 and contains the point (-8,6).

The equation of the parallel line is = (type an equation)

6. Fine the equation of the line with the given properties. Express the equation in general form or slope-intercept form.

Perpendicular to the line 5x+y=57; contains the point (10,6)

7. Find the intercepts and use them to graph the equation.

y=3x+3

8. A circle has the equation x2+y2+8x-8y-49=0. Graph the circle using the center (h,k) and radius r. Find the intercepts, if any, of the graph.

9. For the given functions f and g, find the following and state the domain of each result.

f(x) =√x; g(x) =5x-6

a) (f * g) (x) =

b) What is the domain of f * g?

c) What is the domain of f/g?

10. The function f is defined as follows:

f(x) = { x+5 if -4≤ x 1

a) Find the domain of the function

b) Locate any intercepts

c) graph the function

11. Graph the quadratic function. Give the vertex, axis of symmetry, domain, and range.

f(x) = x2-10x+22

a)The vertex is= (type an ordered pair)

b) The axis of symmetry is x=

c) The domain is = (type your answer in interval notation)

d) The range is = (type your answer in interval notation)

12. Answer the following for the given quadratic function.

f(x) = x2+12x+36

a) What is the vertex (h,k) of f?

b) What is the axis of symmetry?

c) What are the intercepts? (type an ordered pair)

d) on what level is f increasing?

e) on what interval is f decreasing?

13. Find the following for the function f(x) = (x+3)2(x-1)2.

a) find the x- and y-intercepts of the polynomial function f.

b) find the power function that the graph of f resembles for large values of |x|.

c) determine the maximum number of turning points on the graph of f.

d) determine the behavior of the graph of f near each x-intercept.

14. Find the slant asymptote of the graph of the rational function and use the slant asymptote to graph the rational function.

f(x) = x2+x-2/x-5

a) find the slant asymptote of the f

b) Determine the symmetry of the graph of f

c) find the y intercept

d) find the x intercept

e) find the vertical asymptote

f) find the horizontal asymptote

15. solve the polynomial inequality and graph the solution set on a real number line.

x3+x2+49x+49<0

The solution set is = (type answer is interval notation)

16. solve the following inequality.

x2(9+x)(x-5)/(x+2)(x-2) ≥0

what is the solution =

17. Use the rational zeros theorem to find all the real zeros of the polynomial function. use the zeros to factor f over the real numbers.

f(x) = 5x3-x2+5x-1

a) what are the real zeros of f?

b) use the real zeros to factor f (use factored form)

18. Use the given zero to find the remaining zeros of the function.

h(x) = x4-18x3+90x2+54x-1215 zero: 6-3i

19. in f(x) = In x, g(x) =e4x, and h(x) - x5, find the following.

a) (f * g)(x). What is the domain of f * g?

a) (g * f)(x). What is the domain of g * f?

a) (f * h)(x). What is the domain of f * h?

20. Consider the functions f(x) = 7x+4/x+6 and g(x) = 6x-4/7-x

a) find f(g(x))

b) find g(f(x))

c) determine whether the functions f and g are inverses of each other.

21. The function f(x) = 8x/3x-7 is one to one. Find the inverse and check your answer.

f-1(x) =

22. solve the following exponential equation.

ex-e-x/2 =4

a) what is the exact answer.

b) what is the answer approximated to 3 decimal places?

23. Solve the following logarithmic equation.

log (5x+3)=1 + log (x-1)

1. Solve the following equation for x:

log2(x + 7) + log2(x + 8) = 1

Note: Be sure to plug any solutions back into the original equation; if they don't work (e.g., you're forced to take the logarithm of a negative number, which is not possible), discard that solution.

2. The volume V(in cubic meters) of a hot-air balloon is given by V = (4/3) * Pi * R^3.

If the radius r is increasing with time t (in seconds) according to the formula r(t) = 2/3*t^3 , t ≥ 0, find the volume V as a function of the time t.

Solution: V(r(t))=V(2/3*t^3)=4/3*pi(2/3*t^3)^3=4/3*pi(8/27*t^9)=32/81*pi*t^9

3. Find the distance d(P1,P2) between the points P1 and P2.

P1=(1,-3);P2=(2,6)

4. Find the midpoint of the line segment joining the points P1 and P2.

P1=(1,2); P2=(-4,0)

5. Find the equation of a line that is parallel to the line x = 14 and contains the point (-8,6).

The equation of the parallel line is = (type an equation)

6. Fine the equation of the line with the given properties. Express the equation in general form or slope-intercept form.

Perpendicular to the line 5x+y=57; contains the point (10,6)

7. Find the intercepts and use them to graph the equation.

y=3x+3

8. A circle has the equation x2+y2+8x-8y-49=0. Graph the circle using the center (h,k) and radius r. Find the intercepts, if any, of the graph.

9. For the given functions f and g, find the following and state the domain of each result.

f(x) =√x; g(x) =5x-6

a) (f * g) (x) =

b) What is the domain of f * g?

c) What is the domain of f/g?

10. The function f is defined as follows:

f(x) = { x+5 if -4≤ x 1

a) Find the domain of the function

b) Locate any intercepts

c) graph the function

11. Graph the quadratic function. Give the vertex, axis of symmetry, domain, and range.

f(x) = x2-10x+22

a)The vertex is= (type an ordered pair)

b) The axis of symmetry is x=

c) The domain is = (type your answer in interval notation)

d) The range is = (type your answer in interval notation)

12. Answer the following for the given quadratic function.

f(x) = x2+12x+36

a) What is the vertex (h,k) of f?

b) What is the axis of symmetry?

c) What are the intercepts? (type an ordered pair)

d) on what level is f increasing?

e) on what interval is f decreasing?

13. Find the following for the function f(x) = (x+3)2(x-1)2.

a) find the x- and y-intercepts of the polynomial function f.

b) find the power function that the graph of f resembles for large values of |x|.

c) determine the maximum number of turning points on the graph of f.

d) determine the behavior of the graph of f near each x-intercept.

14. Find the slant asymptote of the graph of the rational function and use the slant asymptote to graph the rational function.

f(x) = x2+x-2/x-5

a) find the slant asymptote of the f

b) Determine the symmetry of the graph of f

c) find the y intercept

d) find the x intercept

e) find the vertical asymptote

f) find the horizontal asymptote

15. solve the polynomial inequality and graph the solution set on a real number line.

x3+x2+49x+49<0

The solution set is = (type answer is interval notation)

16. solve the following inequality.

x2(9+x)(x-5)/(x+2)(x-2) ≥0

what is the solution =

17. Use the rational zeros theorem to find all the real zeros of the polynomial function. use the zeros to factor f over the real numbers.

f(x) = 5x3-x2+5x-1

a) what are the real zeros of f?

b) use the real zeros to factor f (use factored form)

18. Use the given zero to find the remaining zeros of the function.

h(x) = x4-18x3+90x2+54x-1215 zero: 6-3i

19. in f(x) = In x, g(x) =e4x, and h(x) - x5, find the following.

a) (f * g)(x). What is the domain of f * g?

a) (g * f)(x). What is the domain of g * f?

a) (f * h)(x). What is the domain of f * h?

20. Consider the functions f(x) = 7x+4/x+6 and g(x) = 6x-4/7-x

a) find f(g(x))

b) find g(f(x))

c) determine whether the functions f and g are inverses of each other.

21. The function f(x) = 8x/3x-7 is one to one. Find the inverse and check your answer.

f-1(x) =

22. solve the following exponential equation.

ex-e-x/2 =4

a) what is the exact answer.

b) what is the answer approximated to 3 decimal places?

23. Solve the following logarithmic equation.

log (5x+3)=1 + log (x-1)

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