Based on the graph of an exponential function f(x)=b^x, for b >0, describe how you can verify that the output of the function can NEVER be equal to zero.

see the explanation

Step-by-step explanation:

we know that

The equation of a exponential growth function is given by

where

a is the initial value or y-intercept

b is the factor growth (b>0)

In this problem

a=1

so

we know that

The graph of the function has no x-intercept

Remember that the x-intercept of a function is the value of x when the value of the function is equal to zero

That means ----> The output of the function can NEVER be equal to zero

Verify

For f(x)=0

Apply log both sides

Remember that

log 0 is undefined. It's not a real number, because you can never get zero by raising anything to the power of anything else.

it would rotate about 180 degrees.

1. when "a" is a root of a polynomial, (x -a) is a factor of it. for the three roots given, the factors of the desired polynomial are (x +2)(x -1/2)(x -3).

in order to make the leading coefficient be negative, we need to multiply this product by a negative number. any negative number will do, but we choose a small (magnitude) value that will eliminate the fraction: -2.

then

f(x) = -2(x +2)(x -1/2)(x -3) = -(x +2)(2x -1)(x -3)

= -(2x² +3x -2)(x -3)

= -(2x³ -3x² -11x +6)

f(x) = -2x³ +3x² +11x -6

2. a graph created by the desmos on-line graphing calculator is shown, and the zeros are highlighted.

3. as indicated in part 1, the multiplier of this equation can be anything and the zeros will remain the same. you want a negative leading coefficient, so the "anything" is restricted to any of the infinite number of numbers that will make that be the case.