Question

Which of the following is true about the exponential function below. Check all that apply

Answer 1

Let us assume we are comparing these function to the function

$z(x)=2^x$

We can tell that function y will have an horizontal asymptote at y=6, because

$\lim _{x\to-\infty}\frac{3}{5}2^{x-7}+6=0+6=6^{}$

So we can discard the first option.

The function y is indeed displaced upward by 6 compared to z, but the horizontal displacement is 7 to the right since y(x)=z(x-7). Thus the second option is incorrect.

Since

$\frac{3}{5}2^x=\frac{3}{5}^{}z(x)$

We can confirm that the function is compressed by a factor of 3/5.

It is not a reflection, and as we discussed earlier when calculating the limit of y as x tends to minus infinity, the domain is all the real numbers, but the function is always greater than 6, so its range is y >6.

Thus, the correct options are 3 and 5. Here's a graph of both functions to confirm our results: