Analytical approach to a quadratics problem

Question

I'm a bit rusty on functions and this exercise got me thinking quite a bit.

The function $y=x$ is tangent to the graph of a certain $g$ function in $x=0$.

Function $g$ can be defined as:

• A) $g(x)=x^2+x$

• B) $g(x)=x^2+x+1$

• C) $g(x)=x^2+2x$

• D) $g(x)=x^2+2x+1$

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Solving the problem graphically I know the only possible answer is A but how would I approach it from an analytical view?

I know that if $y=x$ and that if the function is tanget at $x=0$, $g(0)=0$. On that criteria I can fit both A and C since $g(0)=1$ for B and D.

What am I missing from the logic?

Answer 1

Your approach so far is correct. Yet remember that $y = x$ is tangential to $g(x)$ in $x = 0$. This means that $g'(0) = 1$, since the slope of $y = x$ is $1$ for all $x$. As such, only option A) can be correct, for only A) fulfills the condition $g'(0) = 1$ (of A) and C), having already ruled out B) and D) with $g(0) = 0$)

Answer 2

The derivative of a function at a point is equal to the slope of the tangent at that point. Since the slope of $y=x$ is $1$ you have to check which function satisfies $g'(0)=1$ (together with $g(0)=0,$ of course).