Describe and fine the transformation of two given functions. Then describe the effect on the domain, range, x ntercept, y intercept m, and the asymptote

Question

F(x)=10^x transformed to g(x)=10^(x+1) and h(x)=10^(x-1)

Answer 1

Hi, Shaun.

Given a function f(x), any function of the form f(x - h) has a graph that is a horizontal translation of the graph of f(x). The translation is h units to the right when h > 0 and |h| units to the left when h < 0.

So, in the case of h(x) = 10^{x - 1}, its graph is the graph of f(x) = 10^x translated 1 unit to the right. Shifting the graph horizontally does not affect the domain or range of the function, nor does it affect the graph's asymptote. It does affect the graph's y-intercept because h(0) = 10^{0 - 1} = 10^-1 = 0.1 while f(0) = 10⁰ = 1. Neither function's graph has an x-intercept because neither graph crosses the x-axis.

In the case of g(x) = 10^{x + 1}, you may want to rewrite the function's rule as g(x) = 10^{x - (-1)} in order to recognize that the graph of g(x) is the graph of f(x) = 10^x translated 1 unit to the left. Again, the domain, range, and asymptote do not change. The y-intercept changes because g(0) = 10^{0 + 1} = 10 while f(0) = 10⁰ = 1. Again, neither function's graph has an x-intercept.

I believe that I've discussed all the characteristics called for in the problem, but the problem statement includes "y intercept m", and I don't know what the "m" is doing there. I assume it's just a stray character.

Doug