# Problems from 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. $$\displaystyle g(3)=\frac{1}{64}$$
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### Development:

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 $$Im (f) = (0,+\infty)$$.

### Solution:

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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