Mathematics

Problem

Let

f(x)=a\,b^{x}

be an exponential function satisfying the following conditions: • The point

(0,a)

lies on the graph. • The point

(2,45)

lies on the graph. • Both

a

and

b

are positive integers, with

b>1

and

1\le a\le 10

. a. (i) Show that

b^{2}

must be a factor of

45

and hence determine the possible integer value of

b

. b. (ii) For the value of

b

obtained, find the corresponding value of

a

. c. (iii) Using the values of

a

and

b

, calculate

f(4)

Intuitive explanation

The key is recognizing that $f(0) = a$ directly gives you one parameter, then using the second point to find the other. Since $f(2) = ab^2 = 45$ , you need to find which pair of positive integers $(a,b)$ with $b > 1$ and $a \le 10$ multiply to give 45 when $b$ is squared. Factor 45 and test which factorizations work—this is much faster than guessing.

Intuitive explanation

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