Q1: A linear function is given.
g(x) = −5x + 9
(a) Find the average rate of change of the function between
x = a and x = a + h.
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Q2: A function f is given, and the indicated transformation is applied to its graph. Write the equation for the final transformed graph.
f(x) = √x; shift 3 units to the left
y =
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Q3: Find f ∘ g ∘ h.
f(x) = x − 6, g(x) = √x, h(x) = x − 6
Q1: as always, it is just the slope of the line joining the two points on the graph:
[g(a+h) - g(a)]/h = [(-5(a+h)+9) - (-5a+9)]/h = -5h/h = -5
Q2: y = √(x+3)
review shifting
Q3: f∘g = f(g) = g-6 = √x - 6
f∘h = h-6 = x-6 - 6 = x-12
g∘h = √h = √(x-6)
(f∘g)∘h = √h - 6 = √(x-6) - 6
f∘(g∘h) = √(x-6) - 6
Note that composition is associative so f∘g∘h is unambiguous
thanks for the answers but I wasn't able to understand Question 1 answer can u please explain it?
review slopes. The slope is ∆y/∆x
The average rate of change is the constant slope between two points on the graph. The graph may curve, so the slope will change from point to point. But the slope of the secant line is the constant rate of change that accomplishes the same thing.
A1: To find the average rate of change of a function between two points, we need to calculate the slope of the line passing through those two points. In this case, the points are (x=a, g(a)) and (x=a+h, g(a+h)).
Given the linear function g(x) = -5x + 9, we can substitute the values of x to find the corresponding y coordinates.
The point (x=a, g(a)) on the graph of g(x) is (a, -5a + 9), and the point (x=a+h, g(a+h)) is (a+h, -5(a+h) + 9).
The slope of a line passing through two points, (x1, y1) and (x2, y2), is given by the formula:
slope = (y2 - y1) / (x2 - x1)
So, the average rate of change of g(x) between x=a and x=a+h can be calculated as follows:
average rate of change = (g(a+h) - g(a)) / (a+h - a)
average rate of change = (-5(a+h) + 9 - (-5a + 9)) / (a+h - a)
average rate of change = (-5a - 5h + 9 + 5a - 9) / (a+h - a)
average rate of change = (-5h) / (h)
average rate of change = -5
Therefore, the average rate of change of the function g(x) between x = a and x = a + h is -5.
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A2: To write the equation for the final transformed graph, we need to apply the given transformation to the original function f(x) = √x.
The transformation "shift 3 units to the left" means we need to subtract 3 from the x-coordinate of each point on the graph of f(x).
Starting with the original function f(x) = √x, we can rewrite it as f(x) = √(x + 0) to highlight the x-coordinate.
Applying the shift to the left by 3 units, we subtract 3 from the x-coordinate:
f(x) = √(x + 0 - 3)
f(x) = √(x - 3)
Therefore, the equation for the final transformed graph is y = √(x - 3).
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A3: To find f ∘ g ∘ h, we need to perform the composition of functions in the given order.
Given the functions f(x) = x - 6, g(x) = √x, and h(x) = x - 6, we substitute the expression for g(x) and h(x) into f(x).
f(g(h(x))) = f(g(x - 6))
= f(√(x - 6))
= √(x - 6) - 6
Therefore, f ∘ g ∘ h = √(x - 6) - 6.