An exponential function is a nonlinear function that can be written in the following form, where $a\ne 0,$ $b>0,$ and $b\ne 1.$ As the independent variable $x$ changes by a constant amount, the dependent variable $y$ multiplied by a constant factor. Therefore, consecutive $y\text{-}$values form a constant ratio. Here, the coefficient $a$ is the $y\text{-}$intercept, which is sometimes referred to as the initial value. The base $b$ can be interpreted as the constant factor. The graph of an exponential function depends on the values of $a$ and $b.$

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Why

Why $a\ne 0?$

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If the coefficient $a$ is $0,$ the function becomes a horizontal line.

$$\begin{array}{c}y=0\cdot b^x\Rightarrow y=0\end{array}$$

This is a line along the $x\text{-}$axis and, therefore, is a linear relation. This means that if $a=0,$ then the function is not exponential.

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Why

Why $b>0$ and $b\ne 1?$

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If the base $b$ is negative, the function gives undefined results for certain $x\text{-}$values. For example, since $b^{1/2}=\sqrt{b},$ a negative value for $b$ would yield non-real values for $x=\frac{1}{2}.$ Hence, a condition on $b$ is needed.

$$\begin{array}{c}b\ge 0\end{array}$$

However, if $b=0$ or $b=1$, the function becomes a horizontal line.

$$\begin{array}{ccc}y=a\cdot 0^x& & y=a\cdot 1^x\\ \Downarrow & \text{ and }& \Downarrow \\ y=0& & y=a\end{array}$$

Therefore, $b$ cannot be equal to $0$ nor $1.$

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Further explanation about a difficult or interesting topic.