An exponential function is given by the equation y=3x. Using the points xx and x+1x+1, show that the y-values increase by a factor of 3 between any two points separated by x2−x1=1.(4 points)

Here we have the function:

y = f(x) = 3^x

Using the values:

x and (x + 1)

We need to find that the y-value increases by a factor of 3.

So we need to prove that:

f(x + 1) = 3*f(x).

Or we can see the quotient:

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

Here we can find the values:

f(x + 1) = y = 3^(x + 1)

f(x) = y' = 3^x

If we take the quotient, we get:

Here we can use the properties:

Using these in the quotient equation we get:

Then:

So we found that the y-value increases by a factor of 3 between any two points x₂ and x₁ such that: x₂ - x₁ = 1.

answer: yes, since the line has a steeper rate of change than the exponential the y-values will be greater and they will only intersect one time.