Exercise 61 Page 280

We are asked to show that when x increases by a constant amount k, the quotient f(x+k)f(x) is always the same, regardless the value of x. We are considering that f(x) is an exponential function. f(x) = ab^xTo calculate the quotient, we first need to evaluate the function at x = x + k. f(x) = ab^x ⇒ f( x+k) = ab^(x+k) Now we can continue by calculating the quotient.

f(x+k)/f(x)

SubstituteII

f(x)= ab^x, f(x+k)= ab^(x+k)

ab^(x+k)/ab^x

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Simplify

SimpQuot

Simplify quotient

b^(x+k)/b^x

DivPow

a^m/a^n= a^(m-n)

b^(x+k-x)

SubTerm

Subtract term

b^k

As we can see, the value of the quotient does not depend on x at all. Furthermore, since b and k are constants, the result b^k is constant as well.