The graph of a function h is given. y -6-5-4-3-2 654321 23456 (a) Find h(-2), h(0), h(2), and h(3). h(-2) = 2 h(0) = 0 h(2) = 4 h(3) = 5 (b) Find the domain and range of h. (Enter your answers using interval notation.) domain range (c) Find the values of x for which h(x) = 4. (Enter your answers as a comma-separated list.) x = -3,4 (d) Find the values of x for which h(x) ? 4. (-3, 2) -3, 2, 4 [-3, 2] and 4 [2, 4] and -3 (2, 4) (e) Find the net change in h between x = -3 and x = 3.
The Correct Answer and Explanation is:
(a) Evaluating Specific Points
You are asked to find values of the function h(x)h(x)h(x) at specific xxx-values using the graph.
From the graph:
- h(−2)=2h(-2) = 2h(−2)=2
- h(0)=0h(0) = 0h(0)=0
- h(2)=4h(2) = 4h(2)=4
- h(3)=5h(3) = 5h(3)=5
✅ All values are correctly filled in.
(b) Domain and Range
Domain: The domain is the set of all xxx-values for which the function is defined. From the graph, the function starts at x=−4x = -4x=−4 and ends at x=5x = 5x=5. So:
- Domain = [−4,5][ -4, 5 ][−4,5]
Range: The range is the set of all yyy-values the function takes. From the graph, the lowest value of h(x)h(x)h(x) is 0 (at x=0x = 0x=0), and the highest is 6 (at x=4x = 4x=4):
- Range = [0,6][ 0, 6 ][0,6]
❌ The previous answer was incorrect. Correct domain: [−4,5][ -4, 5 ][−4,5], range: [0,6][ 0, 6 ][0,6]
(c) Find xxx such that h(x)=4h(x) = 4h(x)=4
Looking at the graph, h(x)=4h(x) = 4h(x)=4 at two points:
- x=2x = 2x=2
- x=5x = 5x=5
❌ The answer “-3, 4” is incorrect.
✅ Correct answer: x=2,5x = 2, 5x=2,5
(d) Find values of xxx such that h(x)≤4h(x) \leq 4h(x)≤4
From the graph, h(x)≤4h(x) \leq 4h(x)≤4 from:
- x=−3x = -3x=−3 to x=2x = 2x=2 (inclusive),
- and again at x=4x = 4x=4 where h(4)=4h(4) = 4h(4)=4.
✅ The correct interval is:
−3,2-3, 2−3,2 and x=4x = 4x=4
So the correct multiple-choice answer is:
−3,2-3, 2−3,2 and 4
(e) Net Change from x=−3x = -3x=−3 to x=3x = 3x=3
- h(−3)=4h(-3) = 4h(−3)=4
- h(3)=5h(3) = 5h(3)=5
Net change = h(3)−h(−3)=5−4=1h(3) – h(-3) = 5 – 4 = 1h(3)−h(−3)=5−4=1
✅ Correct answer: 1
Explanation
This problem tests key concepts in reading and interpreting graphs of functions. First, evaluating a function at given points simply involves identifying the y-values at specified x-values on the graph. In (a), locating each x-value on the graph and identifying its corresponding height yields the values for h(x)h(x)h(x).
In (b), understanding the domain and range involves recognizing the horizontal and vertical extent of the graph. The domain corresponds to the leftmost to rightmost x-values for which the graph exists, while the range reflects the lowest to highest y-values. Mistakes here often stem from overlooking endpoints or misinterpreting open versus closed intervals.
Part (c) requires identifying where the graph intersects a horizontal line y=4y = 4y=4. This involves visually locating points on the graph where the height is 4 and reading the corresponding x-values. This question challenges your ability to link function values to x-coordinates and avoid guessing based on symmetry or estimation.
Part (d) examines your understanding of inequalities with functions. You’re not just identifying points where h(x)=4h(x) = 4h(x)=4, but also where the function lies below that level. This tests the ability to read intervals where the graph is beneath or touches a horizontal line.
Finally, (e) asks for the net change, calculated as the difference in output values over an interval. This is a fundamental concept in understanding rates of change and comparing function behavior across intervals. Altogether, this exercise blends graphical interpretation with functional reasoning.
