# Which is the correct answer?

## A function, f, passes through the points (1,1), (2,7) and (3,25). A function, g, passes through the points (1,36), (2,43) and (3,50). Select the correct answer. As the value of x increases, the value of f(x) will never exceed the value of g(x). As the value of x increases, the value of f(x) will eventually exceed the value of g(x). As the value of x increases, the values of f(x) and g(x) remain constant. As the value of x increases, the value of f(x) and the value of g(x) both approach 100.

Jun 22, 2018

I think it is likely that the intended answer was
"As the value of x increases, the value of f(x) will eventual exceed the value of g(x)"
however...

#### Explanation:

Without any information about the nature of $f \left(x\right)$ and $g \left(x\right)$ this question can not be definitively answered. 3 points are not enough to define a function (unless, for example,m we know that the function is a polynomial of degree 2 or less).

From the given information, it seems that $f \left(x\right)$ is likely intended to increase by some non-linear factor, whereas $g \left(x\right)$ is likely intended to be linear.

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If $f \left(x\right)$ and $g \left(x\right)$ are assumed to be continuous polynomial functions with minimal degrees to satisfy the given data,
then
$\textcolor{w h i t e}{\text{XXX}} f \left(x\right) = 6 {x}^{2} - 12 x + 7$
and
$\textcolor{w h i t e}{\text{XXX}} g \left(x\right) = 7 x + 29$
From the graphs we can see that in this case for large values of $x$, $f \left(x\right) > g \left(x\right)$ However (removing the requirement that $f \left(x\right)$ have minimal degree)
$\textcolor{w h i t e}{\text{XXX}} f \left(x\right) = - 6 {x}^{3} + 42 {x}^{2} - 78 x + 43$
also fits the given data values, but the graphs now look like: and except of small values of $x$, $f \left(x\right) < g \left(x\right)$

Similarly, if $g \left(x\right)$ is not of minimal degree... and if we remove the assumption of continuity then anything is possible.