Question

I'm given f (x) = 2x + 3 and g (x) = x squared + 1 and I need to find (gof) (x) = g (f(x)). How do I go about solving this equation?

Answer 1

\[\large \color{royalblue}{g(x)=x^2+1}\]
\[\large \color{orangered}{f(x)=2x+3}\]
Composition of functions.
So what we're going to do is, we're going to take the orange function, and plug it into the blue function.

Let's think for a moment about what happens when you plug values in for x, into the g function.

\[\large \color{royalblue}{g(\color{green}{x})=\color{green}{x}^2+1}\]\[\large \color{royalblue}{g(\color{green}{2})=\color{green}{2}^2+1}\]

We replace every x with the given value, right?
We're going to be doing the same thing here, but instead of plugging in 2, we'll be plugging in f(x) to replace our x.

Answer 2

so now it's g(2) = 5

Answer 3

In the example? Yes good :)
That's just an example though.

Answer 4

oh ok

Answer 5

\[\large \color{royalblue}{g(\color{green}{x})=\color{green}{x}^2+1}\]
So let's plug f(x) in place of our green x.

\[\large \color{royalblue}{g(\color{orangered}{f(x)})=\color{orangered}{f(x)}^2+1}\]

Answer 6

It looks a little messy I know, but understand what we're doing?
Instead of plugging a number in for x, we're plugging in a function.

Answer 7

nice use of color @zepdrix

Answer 8

Ya it's starting to make a little sense

Answer 9

\[\large \color{royalblue}{g(\color{orangered}{f(x)})=\color{orangered}{f(x)}^2+1}\]

Let's remember what our f(x) was, we'll be wanting to plug that in for f(x).\[\large \color{orangered}{f(x)=2x+3}\]

Plugging this in gives us,\[\large \color{royalblue}{g(\color{orangered}{f(x)})=\color{orangered}{(2x+3)}^2+1}\]

Answer 10

So, now would it be g (f (x) ) = 4x squared + 9 + 1 ?

Answer 11

Woops you didn't expand out the square properly.

Remember that \(\large (2x+3)^2=(2x+3)(2x+3)\)
You should get 4 terms coming out of that multiplication in total - two of which you can combine.

Answer 12

Oh right...it should be 4x squared + 12x + 9

Answer 13

Yay good job \c:/

Answer 14

So, am I done with the problem, or is there still more to do?

Answer 15

Ah sorry got distracted. You are indeed done.

\[\large (g \;o\;f)(x)=g(f(x))=4x^2+12x+10\]

Answer 16

Alright thanks.

I just did another problem which required me to find the equation of a line going through points (-2,3) and (5,6). Would I use the distance formula for this one?

Answer 17

no use find the slope first, then use y=mx+b to find b
sub your m and b values into y=mx+b

Answer 18

So, use the midpoint formula?

Answer 19

no, y=mx+b you should close out this problem and post your second problem as a new one

Answer 20

I figured it out