## Precalculus Practice Exam

Part 1 Test 5 Time: 1 hour

1. Given the two points (6,-7) and (3,4), find the following:
• The distance between the two points.

Answer: The distance is .

• The midpoint of the two points.

Answer: The midpoint is .

• An equation of the line between the two points.

Answer: An equation of the line is .

• An equation of the circle whose diameter is the line between the two points.

Answer: An equation of the circle is

2. On inverse functions:
• When does a function have an inverse?

Answer: A function has an inverse only when it is one-to-one, which means that no two elements in the domain of f(x) correspond to the same element in the range of f(x).

• What is the relationship between the domain and range of a function and the domain and range of its inverse?

Answer: The domain of f(x) is the range of its inverse, and the range of f(x) is the domain of its inverse.

• The graphs of a function and its inverse are symmetric to what line?

Answer: They are symmetric to the line y=x.

• Restrict the domain of the function so that its inverse exists.

Answer: There is more than one possible answer. The biggest intervals which will work are either , or .

• If f(x) has an inverse, find .

Answer: .

3. Given the functions , and , find the following.
• Find (f+g)(x).

Answer:

• Find (fg)(x).

Answer:

• Find f(g(x)).

Answer: .

4. How do the following graphs differ from the graph of the function f(x)?
Describe the difference in words.
• f(x)+3.

Answer: It is shifted up vertically by 3 units.

• f(x-4).

Answer: It is shifted to the right by 4 units.

• f(x+5)+6.

Answer: It is shifted up vertically by 6 units and shifted to the left by 5 units.

• -f(x).

Answer: It is reflected across the x-axis.

• 2f(x).

Answer: f(x) is stretched vertically: all values of y are doubled.

5. Concerning the following graph of f(x):
• Approximate the interval(s) where the function is increasing?

Answer: It is increasing on the intervals .

• Approximate the interval(s) where the function is decreasing?

Answer: It is decreasing on the interval [0,2].

6. Solve the inequality

Answer: The inequality is only true when x=4, and when x=-5.

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Thu Jun 26 15:11:07 MDT 1997

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