Let g(x) = -2x + 8 and h(x) = 3x + 4. For what value of x is composition (g o h)(x) = (h o g)(x). If no such value exists, type in DNE in the space provided.

Calculate the two separate functions (g o h)(x) and (h o g)(x).

Some textbooks would write them as
g{f(x)] and f{g(x)}

g{f(x)} = -2(3x+4)+8 = -6 x
f{g(x)} = 3(-2x+8) +4 = -6 x +28

Then set them equal and solve for x.
DNE in this case.

To find the composition (g o h)(x), we substitute h(x) into g(x).

So, (g o h)(x) = g(h(x)), which means we replace x in g(x) with h(x).

First, let's find h(x):
h(x) = 3x + 4

Next, substitute h(x) into g(x):
g(h(x)) = g(3x + 4)

Now, let's evaluate g(3x + 4):
g(x) = -2x + 8
Replacing x with (3x + 4), we get:
g(3x + 4) = -2(3x + 4) + 8

Simplifying this expression:
g(3x + 4) = -6x - 8 + 8
g(3x + 4) = -6x

Now, let's find the composition (h o g)(x), which is h(g(x)).
To do this, we substitute g(x) into h(x):

First, let's find g(x):
g(x) = -2x + 8

Next, substitute g(x) into h(x):
h(g(x)) = h(-2x + 8)

Now, let's evaluate h(-2x + 8):
h(x) = 3x + 4
Replacing x with (-2x + 8), we get:
h(-2x + 8) = 3(-2x + 8) + 4

Simplifying this expression:
h(-2x + 8) = -6x + 24 + 4
h(-2x + 8) = -6x + 28

To find the value of x for which (g o h)(x) = (h o g)(x), we equate the two compositions:

-6x = -6x + 28

Simplifying this equation:
Subtracting -6x from both sides:
0 = 28

The equation 0 = 28 is not true for any value of x, which means there is no value of x for which (g o h)(x) = (h o g)(x). Hence, the answer is DNE (does not exist).