College Algebra Help

[guardgirl08]

On my homework I have the question "Given the functions g(x)=square root of x+7 and h(x)= 1/2x-3. find g(h(x)) and h(g(x))."
I kinda have an idea on how to start this for example: square root of 1/2x-3 +7. Am I going in the right direction?

 

[JeffM]

On my homework I have the question "Given the functions g(x)=square root of x+7 and h(x)= 1/2x-3. find g(h(x)) and h(g(x))."
I kinda have an idea on how to start this for example: square root of 1/2x-3 +7. Am I going in the right direction?

Short answer is "Yes."

Long answer follows.

First, you started out right for this site. Most people here will not give answers; instead they will provide guidance or confirmation, which is what you asked for. Say what the question is, TELL US WHAT WORK YOU HAVE ALREADY DONE, and ask for either guidance or confirmation.

Second, you need to understand order of operations to avoid ambiguity. See http://www.freemathhelp.com/order-of-operations.html. You should write
((1/(2x-3))+7)^(1/2) rather than square root of 1/2x-3 + 7. The ^ (shift key plus the 6) means exponentiation.

Third, g(h(x)) and h(g(x) are compositions of functions. They are sometimes easier to understand if you "substitute variables" (although this is never necessary.)
Let u = g(x) = (x + 7)^(1/2).
Let v = h(x) = (1/(2x - 3)).
Let w = g(h(x)) = g(v) = (v + 7)^(1/2) = ((1/(2x - 3)) + 7)^(1/2), which is what you got.
Let y = h(g(x)) = h(u) = ???

Can you finish?

Is it clear now?

 

[mmm4444bot]

h(x) = 1/2x - 3

As you've typed it, function h first divides its input by 2 and then reduces the result by 3.

So we have:

g(h(x)) = sqrt(1/2x - 3 + 7) = sqrt(1/2x + 4)

Perhaps h(x) is defined differently, in your exercise?

Here is how we use grouping symbols to change the Order of Operations in a typed expression.


Typing 1/2x - 3 means:

\(\displaystyle \frac{1}{2} x - 3\)


Typing 1/(2x) - 3 means:

\(\displaystyle \frac{1}{2x} - 3\)


Typing 1/(2x - 3) means:

\(\displaystyle \frac{1}{2x - 3}\)