What kind of function is this?

Algebra Level 3

A function $f(x)$ has following traits:

$\begin{aligned} f(1) &=78, \\ f(2)&=7878,\\ f(3)&=787878,\\ f(4)&=78787878,\end{aligned}$

and so on. Given that the function can be written as only one of the following expressions:

$\begin{aligned} y&=ax^2+bx+c\\ y&=ab^x+c\\ y&=ax^b+c\\ y&=a\log_b(cx) \end{aligned}$

where $a$ , $b$ , and $c$ are constants. Which one is the correct expression?

y=a\log_bcx

y=ax^b+c

y=ax^2+bx+c

y=ab^x+c

1 solution

Chew-Seong Cheong
Mar 20, 2018

Let $f(0) = 0$ , we note that for positive integers $k$ :

$\begin{aligned} f(k) & = 78(100^{k-1})+f(k-1) \\ f(k) - f(k-1) & = 78(100^{k-1}) \\ \sum_{k=1}^n \left(f(k) - f(k-1) \right) & = \sum_{k=1}^n 78(100^{k-1}) \\ f(n) & = \frac {78(100^n - 1)}{100-1} & \small \color{#3D99F6} \text{Replace }n \text{ with }x \\ \implies f(x) & = \frac {26}{33}\cdot 100^x - \frac {26}{33} \\ & = \boxed{ab^x+c} \end{aligned}$

What kind of function is this?

1 solution

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