Composite Function Homework: Is My Solution Correct?

[Precursor]

Homework Statement

The attempt at a solution

$$g(f(x)) = h(x)$$
$$4f(x) + y = 4x - 1$$
$$4x + 16 + y = 4x - 1$$
$$y = -1 - 16$$
$$y = -17$$

so, $$g(x)= 4x + y = 4x - 17$$

Is this the correct way of going about this question? I used a guessing approach to this question. Is enough work shown to get full marks? Thanks.

 

[Mark44]

Homework Statement

The attempt at a solution

$$g(f(x)) = h(x)$$

==> g(x + 4) = 4x - 1
==> g(x) = 4(x - 4) - 1 = 4x -16 -1 = 4x - 17
Hence g(x) = 4x - 17
The reasoning behind my second equation above is that g(x + 4) represents a translation of g(x) to the left by 4 units, so to get the graph of g, I need to translate it and the function on the right side by 4 units to the right.

$$4f(x) + y = 4x - 1$$

Maybe you can justify the step above, but I don't see it. If the answer was in the back of the book, a guessing approach isn't worth much credit.

$$4x + 16 + y = 4x - 1$$
$$y = -1 - 16$$
$$y = -17$$

so, $$g(x)= 4x + y = 4x - 17$$

Is this the correct way of going about this question? I used a guessing approach to this question. Is enough work shown to get full marks? Thanks.

 

[Precursor]

Thanks for the help. You cleared it up for me.

 

[HallsofIvy]

==> g(x + 4) = 4x - 1
==> g(x) = 4(x - 4) - 1 = 4x -16 -1 = 4x - 17
Hence g(x) = 4x - 17
The reasoning behind my second equation above is that g(x + 4) represents a translation of g(x) to the left by 4 units, so to get the graph of g, I need to translate it and the function on the right side by 4 units to the right.

Another way to do this. Since g(f(x))= g(x+ 4)= 4x- 1, let y= x+ 4. Then x= y- 4 so 4x-1= 4(y- 4)- 1= 4y- 17. g(x+4)= g(y)= 4y- 17 and, since the "y" is just a "placeholder", g(x)= 4x- 17.

Maybe you can justify the step above, but I don't see it. If the answer was in the back of the book, a guessing approach isn't worth much credit.