Show that $b^{x}$ is bijective

Question

Show that a function defined by $f(x) = b^{x} : \mathbb R \rightarrow (0,\infty)$, where $b > 1$ is a bijective function and therefore invertable. The parenthesis around $0, \infty$ means that the target set is created than $0$ but less than $\infty$. Any help is appreciated.

Answer 1

Hint 1: to show a function is bijective, you must show

1. it is one-to-one, i.e. "injective", and
2. it is onto, i.e. "surjective"

Hint 2:

• To show the function is injective start with $f(a) = f(b)$, and use what you know about the function $f$ to reach the conclusion that $a = b$. (Side note: the visual method of doing this is called the "horizontal line test.")
• To show the function is surjective, start with $f(x) = y$, and show that, no matter what $y$ is, you can always solve for $x$.

Hope this helps!

Answer 2

As per your comment, I will further address an answer. Take a look here. They use $e^x$ instead of $b^x$, but if you can prove that $e^x$ is invertible, you can adjust the proof for $b^x$.