Answer

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Hint: We have to first check the continuity of the function in the given interval. Then we have to find the limits of the function at its extreme intervals.

Complete step-by-step answer:

So here we are given a function ${{e}^{x}}$ in [0, 1] and we have to find its range.

First we need to find if this function is continuous in [0, 1].

f (x) = ${{e}^{x}}$

f (0) = ${{e}^{0}}$

= 1

f (1) =${{e}^{1}}$

= e

As the limits exist in the interval and there are no corner points where the function becomes discontinuous, we can say that the function is continuous in the interval [0, 1].

Here the function f(x) = ${{e}^{x}}$ is increasing in nature, which means f(x+1)>f(x).

So f(1) > f(0)

f (0) = ${{e}^{0}}$

= 1

f (1) =${{e}^{1}}$

= e

This is the graph of ${{e}^{x}}$. We can see that the function tends to 0 when x tends to -∞. And when x tends to ∞ then the function ${{e}^{x}}$ also tends to infinity. It intersects the y axis at the point (0,1) which means when the value of x = 0 , then the value of the function is 1.

We can see this is an increasing function and it is continuous in the interval [0, 1].

Therefore the range of the function f(x) = ${{e}^{x}}$ is [0, 1]

Note: We must remember the graphs of ${{e}^{x}}$ and ${{e}^{-x}}$ as they are very much important for boards as well as competitive exams. Where ${{e}^{x}}$ is always increasing in nature and on the other hand ${{e}^{-x}}$ is always decreasing in nature. They are also called exponential graphs, which are used to model populations.

Complete step-by-step answer:

So here we are given a function ${{e}^{x}}$ in [0, 1] and we have to find its range.

First we need to find if this function is continuous in [0, 1].

f (x) = ${{e}^{x}}$

f (0) = ${{e}^{0}}$

= 1

f (1) =${{e}^{1}}$

= e

As the limits exist in the interval and there are no corner points where the function becomes discontinuous, we can say that the function is continuous in the interval [0, 1].

Here the function f(x) = ${{e}^{x}}$ is increasing in nature, which means f(x+1)>f(x).

So f(1) > f(0)

f (0) = ${{e}^{0}}$

= 1

f (1) =${{e}^{1}}$

= e

This is the graph of ${{e}^{x}}$. We can see that the function tends to 0 when x tends to -∞. And when x tends to ∞ then the function ${{e}^{x}}$ also tends to infinity. It intersects the y axis at the point (0,1) which means when the value of x = 0 , then the value of the function is 1.

We can see this is an increasing function and it is continuous in the interval [0, 1].

Therefore the range of the function f(x) = ${{e}^{x}}$ is [0, 1]

Note: We must remember the graphs of ${{e}^{x}}$ and ${{e}^{-x}}$ as they are very much important for boards as well as competitive exams. Where ${{e}^{x}}$ is always increasing in nature and on the other hand ${{e}^{-x}}$ is always decreasing in nature. They are also called exponential graphs, which are used to model populations.

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