Exponential functions

Indicate what is the base of the exponential functions that satisfy the following relations. Indicate, also, its domain and image:

  1. $f(2)=16$
  2. $h(-2)=25$
  3. $\displaystyle g(3)=\frac{1}{64}$

  1. To find the base of the exponential function we raise and solve the following equation: $$x^2=16 \Rightarrow x=4$$ Therefore $4$ is the base of the function, with $Dom (f) = \mathbb{R}$ and $Im (f) = (0,+\infty)$.

  2. We proceed as in the previous case: $$x^{-2}=25 \Rightarrow x^2=\dfrac{1}{25} \Rightarrow x=\dfrac{1}{5}$$ Therefore the base of the function is $\dfrac{1}{5}$, with $Dom (f) = \mathbb{R}$ and $Im (f) = (0,+\infty)$.

  3. We proceed as in the previous case: $$x^3=\dfrac{1}{64} \Rightarrow x=\sqrt[3]{\dfrac{1}{64}} \Rightarrow x=\dfrac{1}{4}$$ Therefore the base of function is $\dfrac{1}{4}$, with $Dom (f) =\mathbb{R}$ and $Im (f) = (0,+\infty)$.

  1. $b=4$, $Dom (f) = \mathbb{R}$, $Im (f) = (0,+\infty)$
  2. $b=\dfrac{1}{5}$, $Dom (f) = \mathbb{R}$, $Im (f) = (0,+\infty)$
  3. $b=\dfrac{1}{4}$, $Dom (f) =\mathbb{R}$, $Im (f) = (0,+\infty)$
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