# Growth of Exponential Functions

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Proposition A:

If [math]f[/math] is an exponential function of the form [math]f(x) = a b^x[/math], [math]a>0, \ b>0 " and " b!=1[/math], and [math]\alpha > 0[/math] is a given constant, then [math](f(x+alpha))/f(x) = c[/math] for all values of [math]x in RR[/math], where [math]c[/math] is a real-valued constant.

If [math]f[/math] is an exponential function of the form [math]f(x) = a b^x[/math], [math]a>0, \ b>0 " and " b!=1[/math], and [math]\alpha > 0[/math] is a given constant, then [math](f(x+alpha))/f(x) = c[/math] for all values of [math]x in RR[/math], where [math]c[/math] is a real-valued constant.

A.

Using the definition of [math]f(x)[/math] in the proposition, rewrite [math]f(x + \alpha) / f(x)[/math], in simplest terms.

- [math]2ab^{x+\alpha}[/math]
- [math]b^{x+2 \alpha}[/math]
- [math]b^alpha[/math]
- [math]\alpha[/math]

B.

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

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

C.

One way to restate Proposition A is as follows: "Exponential functions grow by equal factors over equal intervals." What is the value of the equal factors and length of the equal intervals as represented in Proposition A?

- Equal factors are [math]x_0[/math], equal intervals are [math]c[/math].
- Equal factors are [math]alpha[/math], equal intervals are [math]x[/math].
- Equal factors are [math]alpha[/math], equal intervals are [math]c[/math].
- Equal factors are [math]c[/math], equal intervals are [math]alpha[/math].