Question

Consider two functions: g(x)=20(1.5)x and the function ​f(x)​ shown in the table.

Which statements are true?

Select each correct answer.




​ f(1) is less than g(−1) .

​f(x)​ increases at a faster rate than ​g(x)​ does on the interval (−5, −3) .

​g(x)​ has a greater y-intercept than ​f(x)​ does.

f(x) and ​g(x)​ are both increasing on the interval (−∞, ∞) .
x f(x)
−5 −45
−4 −48
−3 −49
−2 −48
−1 −45
0 −40
1 −33

Answer 1

The correct options are 1 and 3.

Explanation:

The given function is

`g(x)=20(1.5)^x`

1.

From the given table the value of f(x) is -33 at x=1, so

`f(1)=-33`

Substitute x=-1 in g(x), to find the value of g(-1).

`g(-1)=20(1.5)^(-1)=13.33`

Since -33<13.33, therefore​ f(1) is less than g(−1) and statement 1 correct.

2.

From the table it is notices that the value of f(x) is -45 at x=-5 and -49 at x=-3.

The rate of change is

`m=(-49-(-45))/(-3-(-5))=(-4)/(2)=-2`

The function g(x) is a growth function because the exponent 1.5>1, therefore the rate of change is always positive.

So, the rate of change of g(x) is greater than the f(x).

​The statement "f(x)​ increases at a faster rate than ​g(x)​ does on the interval (−5, −3)" is incorrect.

3.

From the given table it is notices that the y-intercept is -40 because the function intersects the y-axis at (0,-40).

Substitute x=0 in g(x).

`g(0)=20(1.5)^0=20`

Since ​g(x)​ has a greater y-intercept than ​f(x)​ does, therefore statement 3 is correct.

4.

From the given table it is noticed that the function f(x) first decreases and after that it increases on the interval
`(-\infty, \infty)`.

The function g(x) is a growth function, therefore the function g(x) increases on the interval
`(-\infty, \infty)`.

Since the function f(x) is not satirically increasing on the interval
`(-\infty, \infty)`, therefore statement 4 is incorrect.

Answer 2

we have
g(x)=20(1.5)x
f(x)

Create a table with a different values of x for g(x)
see the attached table

1 f(1) is less than g(−1) -------------- > is true
f(1)=-33 and g(1)=30 f(1)< g(1)

2 f(x)​ increases at a faster rate than ​g(x)​ does on the interval (−5, −3)
f(x) for x=-5 f(-5)=-45 for x=-3 f(-3)=-49 rate=(-49+45)/(-3+5)=-2
g(x) for x=-5 g(-5)=-150 for x=-3 g(-3)=-90 rate=(-90+150)/(-3+5)=30
is a false statements
because g(x)​ increases at a faster rate than ​f(x)​

3 ​g(x)​ has a greater y-intercept than ​f(x)​ does.
g(x)​ y-intercept is 0
f(x)​ y-intercept is -40
is a true statements
because 0>-40

4 f(x) and ​g(x)​ are both increasing on the interval (−∞, ∞)
is a true statements