Grade 10 Functions — Advanced Practice — Easy

Question 1 of 25Score 0/0Easy

Question 1 of 25: f(x) + k shifts the graph:

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Answer key for parents & teachers (25 questions)
  1. f(x) + k shifts the graph: — Up by k. Adding k outside the function (vertical change) shifts the graph UP by k units.
  2. f(x) − k shifts the graph: — Down by k. Subtracting k outside the function shifts the graph DOWN by k units.
  3. f(x − h) shifts the graph: — Right by h. f(x − h) shifts the graph RIGHT by h. Remember: horizontal shifts work in the OPPOSITE direction of the sign inside.
  4. f(x + h) shifts the graph: — Left by h. f(x + h) shifts the graph LEFT by h. Inside the function, + means left.
  5. −f(x) reflects the graph over: — The x-axis. −f(x) negates all y-values, reflecting the graph over the x-axis.
  6. f(−x) reflects the graph over: — The y-axis. f(−x) replaces x with −x, reflecting the graph over the y-axis.
  7. g(x) = f(x) + 5 compared to f(x) is: — Shifted up 5. Adding 5 outside → vertical shift up 5 units.
  8. (f ∘ g)(x) means: — f(g(x)). (f ∘ g)(x) = f(g(x)) — apply g first, then apply f to the result.
  9. f(x) = x + 2, g(x) = 3x. What is (f ∘ g)(x)? — 3x + 2. (f ∘ g)(x) = f(g(x)) = f(3x) = 3x + 2.
  10. f(x) = x + 2, g(x) = 3x. What is (g ∘ f)(x)? — 3x + 6. (g ∘ f)(x) = g(f(x)) = g(x + 2) = 3(x + 2) = 3x + 6.
  11. Does composition generally commute? (Is f ∘ g = g ∘ f?) — No, composition is generally NOT commutative. Composition is NOT generally commutative. (f ∘ g)(x) ≠ (g ∘ f)(x) in most cases.
  12. f(x) = 2x − 1. Find f⁻¹(x). — f⁻¹(x) = (x + 1)/2. Swap x and y: x = 2y − 1 → y = (x + 1)/2. So f⁻¹(x) = (x + 1)/2.
  13. A function has an inverse function if and only if it is: — One-to-one (passes horizontal line test). A function has an inverse iff it is one-to-one: each y-value is paired with at most one x-value (passes horizontal line test).
  14. f(x) = 2x, g(x) = x + 3. Find (f ∘ g)(5). — 16. g(5) = 8. f(g(5)) = f(8) = 16.
  15. f(x) = 2x, g(x) = x + 3. Find (g ∘ f)(5). — 13. f(5) = 10. g(f(5)) = g(10) = 13.
  16. Which test determines if a function has an inverse? — Horizontal line test. The horizontal line test: if any horizontal line crosses the graph more than once, the function is NOT one-to-one and has no inverse.
  17. A piecewise function uses different formulas for different: — Intervals of the domain. A piecewise function defines different formulas over different parts (intervals) of its domain.
  18. For f(x) = {2x if x < 0; x² if x ≥ 0}, what is f(3)? — 9. x = 3 ≥ 0, so use f(x) = x². f(3) = 9.
  19. For f(x) = {2x if x < 0; x² if x ≥ 0}, what is f(−2)? — −4. x = −2 < 0, so use f(x) = 2x. f(−2) = 2(−2) = −4.
  20. For f(x) = {x + 3 if x ≤ 1; x² − 1 if x > 1}, what is f(0)? — 3. x = 0 ≤ 1, so use f(x) = x + 3. f(0) = 0 + 3 = 3.
  21. The absolute value function |x| is a piecewise function defined as: — x if x ≥ 0; −x if x < 0. |x| = x when x ≥ 0 (positive values stay positive) and |x| = −x when x < 0 (negates negative to make positive).
  22. To evaluate a piecewise function, the first step is to: — Identify which piece's domain the input belongs to. First identify which interval the input x falls in, then use that piece's formula to evaluate.
  23. f(x) = {x + 1 if x ≤ 0; 2x if x > 0}. Find f(0). — 1. x = 0 satisfies x ≤ 0. Use f(x) = x + 1. f(0) = 0 + 1 = 1.
  24. f(x) = {x² if x < 2; 4 if x ≥ 2}. What is f(2)? — 4. x = 2 satisfies x ≥ 2. Use f(x) = 4. f(2) = 4.
  25. f(x) = {−x if x < −1; x + 2 if −1 ≤ x < 2; 4 if x ≥ 2}. How many pieces does this function have? — 3. Three separate conditions define three pieces.