Growth of Exponential Functions

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Proposition A:

If

$f$ is an exponential function of the form

$f(x) = a b^x$ ,

$a>0, \ b>0 " and " b!=1$ , and

$\alpha > 0$ is a given constant, then

$(f(x+alpha))/f(x) = c$ for all values of

$x in RR$ , where

$c$ is a real-valued constant.

Grade 11 Functions and Relations CCSS: HSF-LE.A.1, HSF-LE.A.1a

Using the definition of

$f(x + \alpha) / f(x)$ , in simplest terms.

Grade 11 Functions and Relations CCSS: HSF-LE.A.1, HSF-LE.A.1a

Which of the following reasons best explains why proposition A is true?

Since $\alpha$ is a constant value, the value of $f(x+\alpha)/f(x)$ will also be a constant value.
Because the simplified form of $f(x+\alpha)/f(x)$ is an exponential equation, it is valid for all values of $x in RR$ , and will be constant.
Since the simplified form of $f(x+\alpha)/f(x)$ is independent of $x$ , it will be a constant value.
Because a constant value to the power of a constant value is also a constant, the value $f(x+\alpha)/f(x)$ will also be a constant value.

Grade 11 Functions and Relations CCSS: HSF-LE.A.1, HSF-LE.A.1a