How are the graphs of f(x) = e ^ x - 2 g(x) = ln(x + 2) related if they are each graphed on the same grid.

Answer 1

Question: How are the graphs of f(x) = e^x - 2 and g(x) = ln(x + 2) related if they are each graphed on the same grid?

Answer: We're comparing an exponential function and a logarithmic function. WIth that kind of question, I suspect inverses. The work below is algebraic proof that f(x) and g(x) are inverses of each other. I started with f(x) and changed the f(x) notation to y for convenience, and then switched x and y, and solved for the new y.

f(x) = e^x - 2

y = e^x - 2

x = e^y - 2

x + 2 = e^y

ln(x + 2) = ln(e)^y

ln(x + 2) = y

y = ln(x + 2)

f(x) and g(x) are inverses. If you were to graph these functions on the same plot, you would find that these functions mirror each other, but where's the mirror? It's about the line y = x, and this is a property of inverse functions: they reflect each other about the line y = x. Type the functions into a graphing calculator and experience it for yourself! Mirror, mirror on the wall...