Functions

1. What is the natural real domain of f(x) = 1/(x - 4)?

1. All real x except x = 4
2. x > 4 only
3. x >= 4
4. All real x except x = -4

2. For real x, what is the range of g(x) = x^2 + 5?

1. g(x) > 5
2. g(x) >= 5
3. g(x) <= 5
4. All real values

3. Find the range of h(x) = 1/x when its domain is x >= 2.

1. h(x) >= 1/2
2. 0 <= h(x) <= 1/2
3. 0 < h(x) <= 1/2
4. h(x) < 1/2

4. Which relation does not define y as a function of x?

1. y = 3x - 1
2. y = x^2
3. y = sqrt(x) for x >= 0
4. x = y^2 for real y

5. Given f(x) = 2x - 3 and g(x) = x^2, find gf(x).

1. (2x - 3)^2
2. 2x^2 - 3
3. 4x^2 - 3
4. 2(x - 3)^2

6. Let f(x) = sqrt(x) and g(x) = x - 5. What are fg(x) and its real domain?

1. sqrt(x) - 5, x >= 0
2. sqrt(x - 5), x >= 5
3. sqrt(x - 5), x <= 5
4. x - sqrt(5), all real x

7. Which condition ensures that gf can be formed on the whole stated domain of f?

1. The domain of f equals the domain of g
2. The range of g lies within the domain of f
3. The range of f lies within the domain of g
4. Both functions have the same range

8. Let f(x) = x^2 for -1 <= x <= 3 and g(x) = 1/(x - 4). What is the domain of gf?

1. -1 <= x <= 3
2. -1 <= x < 2
3. 2 < x <= 3
4. -1 <= x <= 3, with x = 2 excluded

9. Which function is one-one on the set of all real numbers?

1. f(x) = 3x - 7
2. f(x) = x^2
3. f(x) = |x|
4. f(x) = (x - 2)^2 + 1

10. The function f(x) = x^2 is restricted to x >= 0. What is f inverse(x)?

1. -sqrt(x), x >= 0
2. sqrt(x), x >= 0
3. x^2, x >= 0
4. +/-sqrt(x), x >= 0

11. If h(x) = 5 - 2x, find h inverse(x).

1. (x - 5)/2
2. 5 - x/2
3. (5 - x)/2
4. 1/(5 - 2x)

12. Let f(x) = (x - 1)^2 + 4 for x <= 1. Which is its inverse?

1. 1 + sqrt(x - 4), x >= 4
2. 1 - sqrt(x + 4), x >= -4
3. -1 - sqrt(x - 4), x >= 4
4. 1 - sqrt(x - 4), x >= 4

13. For f(x) = (3x - 5)/(x + 2), what is f inverse(x)?

1. (2x + 5)/(3 - x)
2. (2x - 5)/(x + 3)
3. (3x + 5)/(2 - x)
4. (5 - 2x)/(x - 3)

14. A one-one function satisfies f(2) = 7. What must be true?

1. f inverse(2) = 7
2. f inverse(7) = 2
3. f(7) = 2
4. f inverse(7) = -2

15. The graph of a one-one function is reflected to obtain the graph of its inverse. What is the mirror line?

1. x = 0
2. y = 0
3. y = x
4. y = -x

16. A one-one function has domain x > 1 and range y < 4. What are the domain and range of its inverse?

1. Domain x > 1, range y < 4
2. Domain x < 1, range y > 4
3. Domain x > 4, range y < 1
4. Domain x < 4, range y > 1

17. How is y = f(x) + 3 obtained from y = f(x)?

1. Translate up 3 units
2. Translate right 3 units
3. Stretch vertically by factor 3
4. Reflect in the x-axis

18. How is y = f(x + 4) obtained from y = f(x)?

1. Translate right 4 units
2. Translate left 4 units
3. Translate up 4 units
4. Stretch horizontally by factor 4

19. Which transformation maps y = f(x) to y = -f(x)?

1. Reflection in y = x
2. Reflection in the y-axis
3. Reflection in the x-axis
4. A horizontal stretch

20. Which transformation maps y = f(x) to y = f(-x)?

1. Reflection in the x-axis
2. Reflection in y = x
3. A vertical stretch
4. Reflection in the y-axis

21. How is y = 3f(x) obtained from y = f(x)?

1. Vertical stretch by factor 3
2. Horizontal stretch by factor 3
3. Translation up 3 units
4. Horizontal scale factor 1/3

22. How is y = f(2x) obtained from y = f(x)?

1. Horizontal stretch by factor 2
2. Horizontal scale factor 1/2
3. Vertical stretch by factor 2
4. Translation left 2 units

23. The point (6, -4) lies on y = f(x). Which point lies on y = -2f(3x)?

1. (18, 8)
2. (2, -8)
3. (2, 8)
4. (18, -8)

24. The point (-3, 4) lies on y = f(x). Which point lies on y = 2f(-(x - 1)) + 5?

1. (-2, 13)
2. (4, 3)
3. (-4, 13)
4. (4, 13)