Consider the function f(x)=2(1.2^x). 1.) This is a.) an exponential growth function  b.) a logarithmic function c.) an exponential decay function d.) a function using natural logarithms 2.) The...

Consider the function f(x)=2(1.2^x).

1.) This is

a.) an exponential growth function 

b.) a logarithmic function

c.) an exponential decay function

d.) a function using natural logarithms

2.) The y-intercept

a.) is at x=1.2

b.) is at y=1.2 

c.) is at y=2

d.) does not exist

e.) none of the above

3.) The x-intercept

a.) is at x=0

b.) is at x≈1.2

c.) is at y=1.2

d.) does not exist

e.) none of the above

The end behavior of this function is best described as:

A.) f(x)->-∞ as x->-∞; f(x)->120 as x->∞

B.) f(x)->0 as x->-∞; f(x)->∞ as x->-∞

C.) f(x)->-∞ as x->0; f(x)->∞ as x->120

D.) f(x)->1.2 as x->-∞; f(x)->-∞ as x->∞

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Borys Shumyatskiy | College Teacher | (Level 3) Associate Educator

Posted on

I think that you mean f(x) = 2*(1.2)^x.

It may be rewritten as f(x) = 2*e^(x*ln(1.2)) because e^(a*b) = e^a*e^b and e^(ln(a)) = a. Note that 1.2>1 and therefore ln(1.2)>0. Now answer your questions:

1) f(x) is of the form a*e^(b*x) with a>0 and b>0. This is an exponential function and it  grows. So the answer is a.) an exponential growth function.

2) the y-intercept of a function f is a point (0, f(0)). Here it is (0, 2*e^0) = (0, 2). The answer is c.) is at y=2.

3) the x-intercept is a root, a point x where f(x) = 0. There are no such points, e^(a) is always positive. The answer is d.) does not exist.

4) The end behavior of this function. I hope you know that
[a*e^(b*x)] --> +∞ when x->+∞ (a>0, b>0)
and
[a*e^(b*x)] --> +0 when x->-∞.

So variants A) and D) are incorrect. C) is incorrect because f is continuous and f(x) --> f(0) as x-->0? and f(0)=2, not -∞.

Variant B) is true if we correct (I believe) a mistake:
B.) f(x)->0 as x->-∞; f(x)->∞ as x->+∞.
(I corrected -∞ to the +∞ at the right).

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