Question

If `f(x) = 2x^2 + 5` and `g(x) = 3x + a`, how do you find a so that the graph of (f o g)(x) crosses the y-axis at 23?

Answer 1

I assume that `(fog)(x)` means `f(g(x))`

Explanation:

First let's find `f(g(x))=2(3x+a)^2+5`, replacing `(3x+a)` into `f()`. We then have `f(g(x))=2(9x^2+6a+a^2)+5=(18x^2+12a+2a^2)+5`.

Now the graph crosses the y-axis when `x=0`, so we must have:

`12a+2a^2+5=23`, that is `12a+2a^2-18=2a^2+12a-18=0`. We must now solve the equation for `a`.

We may first divide the equation by `2` (just to make the calculation slightly easier), so we get `a^2+6a-9=0`, then the roots of the quadratic equation are:

`(-6+-sqrt(36+4*9))/2=(-6+-sqrt(72))/2=(-6+-6sqrt(2))/2`.

Then the two solutions are `a=(-3+3sqrt(2))` and `a=(-3-3sqrt(2))`