Question

Please help me asap!!

8. Compare the function g(x) modeled by the graph and the functionf (x) =
21x + 2) + 3.
True or false. Explain.
a) f(x) has a greater y intercept
b) g(1) > f(-1)
c) f(x) increases at a faster rate than g(x)over interval [0, 3).

Answer 1

a) false

b) true

c) true

Explanation:

Part (a)

The y-intercept of a function is the point when x = 0

⇒ f(0) = 2|0 + 2| + 3

= 2|2| + 3

= 4 + 3

= 7

From inspection of the graph, the y-intercept of g(x) is (0, 7)

Therefore, the y-intercepts of f(x) and g(x) are the same.

"f(x) has a greater y-intercept" → false

Part (b)

From inspection of the graph, g(1) = 6

⇒ f(-1) = 2|-1 + 2| + 3

= 2|1| + 3

= 2 + 3

= 5

Therefore, g(1) > f(-1) → true

Part (c)

The average rate of change of function f(x) over the interval a ≤ x ≤ b is given by:

`(f(b)-f(a))/(b-a)`

From inspection of the graph:

g(0) = 7

g(3) = 4

`\textsf{average rate of change of g(x)} \implies (4-7)/(3-0)=-1`

f(0) = 7 (from part (a))

f(3) = 2|3 + 2| + 3

= 2|5| + 3

= 10 + 3

= 13

`\textsf{average rate of change of f(x)} \implies (13-7)/(3-0)=2`

Function g(x) decreases over [0, 3] at the rate of -1

Function f(x) increases over [0, 3] at the rate of 2

Therefore, "f(x) increases at a faster rate than g(x) over interval [0, 3]" is true

**Please see the attached graph for reference**

The interval [0, 3] is shaded in grey

f(x) is shown in red

The added 2 points are f(-1) and g(1)