Question

f(x) = -4x + 7 and g(x) = 10x - 6. Find f(g(x))

So, lets see... -4x(10x - 6) + 7, right?

Answer 1

No, you replace the x in -4x + 7 with the (10x-6).

Answer 2

What is all this business about replacing x. WHAT X?

Answer 3

f(x) = -4x + 7
you see the x right next to the -4, THAT X.

Answer 4

\[\Large\rm f(\color{royalblue}{x})=-4\color{royalblue}{x}+7\]\[\Large\rm f(\color{royalblue}{g(x)})=-4\color{royalblue}{g(x)}+7\]\[\Large\rm f(\color{royalblue}{g(x)})=-4\color{royalblue}{(10x-6)}+7\]

Answer 5

Yah that x shouldn't next to the -4 anymore, after you've replaced it :o

Answer 6

Oh, now I see, Zepdrix.

So f(g(x)) = -4(10x - 6) + 7
-40x +24 + 7

Which yields.... -40x + 31

What is the next step, your heinous?

Answer 7

That would be as far as you can simplify it :)
See the notation on the left side of the equals?
f(g(x)),

it's still a function of x at this point,
it's not being evaluated at any particular number.

Answer 8

For example if you wanted to evaluate this composition at x=2, you would have,
f(g(2)) = -40(2)+31
f(g(2)) = -49

But that's not what they asked for here :)

Answer 9

What is this "composition" you speak of? And how do I know what I need to "evaluate."

Answer 10

So we started with f(x) and g(x).

\(\Large\rm f(g(x))\) reads -> f of g of x
sometimes we use this notation instead (meaning the same thing):
\(\Large\rm (f \circ g)(x)\) reads -> f composed of g of x

It's a composition of functions, a function within a function.

Answer 11

The question said \(\Large\rm find~f(g(\color{royalblue}{x}))\)

Pay attention to the blue x, if it's a number, then that's when they're asking you to evaluate it at a particular number.

Answer 12

If it's a number, you should end up with a number after doing your calculations.
If it's a variable (like they asked for), you should end up with a function (stuff including x, like we did).

Answer 13

Okay, so since we have "f(g(x))"

We're given "x" which is what we're being asked to find. For example, if we had f(g(1)), we'd look for 1, right?

Answer 14

So now that I have: -40x + 31

I plug that into f(x)

f(-40x + 31) = -4(-40x + 31) + 7

Correct?

Answer 15

If we were asked to find f(g(1)),
we could do the same steps we did,
first finding f(g(x)),

f(g(x)) = -40x + 31

and then plug x=1 into the function to find f(g(1))

f(g(1)) = -40(1) + 31

Answer 16

But we need to plug -40x + 31 into f(x), not 1, right?

Answer 17

What you did was, you plugged f(g(x)) into your original f(x) function.

So what you just found was f(f(g(x))).

Lemme do it in color, maybe it'll be a little more clear.

Answer 18

We found \(\Large\rm f(g(x))\),

and you plugged that whole thing into \(\Large\rm f\left(\color{orangered}{x}\right)=-4(\color{orangered}{x})+7\)

and it gave you
\(\Large\rm f\left(\color{orangered}{f(g(x)}\right)=-4(\color{orangered}{f(g(x)})+7\)

\(\Large\rm f\left(\color{orangered}{f(g(x)}\right)=-4(\color{orangered}{-40x+31})+7\)

Answer 19

That's a little too fancy :O That's not what we want to be doing hehe

Answer 20

Well, what are we trying to do cx

Answer 21

So you're feeling ok with this first step, yes?
Plugging a function into a function?
At least a little bit better?