Why is this not an exponential function?

Question

$$5\times 1.2^x-3$$

Is it simply that it doesn't follow the form "$ca^x$"?

Answer 1

Yes, it can't be written in that form. Notice that function of the form of $ca^x$ doesn't change sign.

But for our function $f(x)=5 \times 1.2^x-3$,

$$\lim_{x \to -\infty}f(x)=-3$$ and $$\lim_{x \to \infty}f(x)=\infty.$$

Answer 2

Yes, the $-3$ is exactly what makes your function not exponential.

Answer 3

It doesn't follow that $\,f(x+y)=f(x)f(y)\,$, which is what makes a function eligible to be even considered as being called exponential. For why not all functions that obey the above are actually exponential, see the answers under Overview of basic facts about Cauchy functional equation.

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[ EDIT ]  If using a definition of exponentials which includes functions of the form $\,\color{red}{c \cdot} a^x\,$, then the equality above needs to be changed to $\,f(0)f(x+y)=f(x)f(y)\,$.