Longtime lurker here, composite function problem

In summary: I'm not quite sure what you mean by "not perfect". It looks great from here.And thanks for the heads up on the "package" nomenclature. I'll try to remember that.This is a very useful package, and I especially appreciate the way that it will accept a formula, and plot the graph for you.Ray.In summary, the conversation discusses finding the function g(x) given the composite function f(g(h(x))) and two other functions, f(x) and h(x). The solution involves systematically substituting h(x) into f(g(h(x))) and solving for g(x). The resulting function is g(x) = sqrt((10-x)/(7-x)) with a range of y ≥
  • #1
Poset
3
0

Homework Statement



Let f(g(h(x))) = 1/(2-x)

Find g(x) if:

f(x) = (x^2) - 1
h(x) = 3x+12. The attempt at a solution

This is what I have:

g(h(x))^2 -1 = 1/(2-x)
g(h(x) = sqrt((3-x)/(2-x))

I'm not sure how to get the h(x) out of this to leave me with just g(x). Please point me in the right direction.

Thank you!
 
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  • #2
The obvious thing to do is to set [itex]y = h(x)[/itex] and find [itex]g(y) = g(h(x))[/itex] in terms of [itex]y[/itex].
 
  • #3
Thank you for your reply. Am I right to assume that the calculations to g(h(x)) I did are correct? And what you are implying is that to get g(x) I need to go backwards from h(x) using the inverse of g(h(x)) and y = h(x) = 3x+1? I'm sorry it's not that obvious to me.
 
  • #4
Your second line isn't far off. Then if you replace x there with y=h-1 you'll get g(y).
 
  • #5
Poset said:

Homework Statement



Let f(g(h(x))) = 1/(2-x)

Find g(x) if:

f(x) = (x^2) - 1
h(x) = 3x+12. The attempt at a solution

This is what I have:

g(h(x))^2 -1 = 1/(2-x)
g(h(x) = sqrt((3-x)/(2-x))

I'm not sure how to get the h(x) out of this to leave me with just g(x). Please point me in the right direction.

Thank you!

You've got:

##g(h(x)) = \sqrt{\frac{3-x}{2-x}}##

Why not take one more step?

##g(3x+1) = \sqrt{\frac{3-x}{2-x}}##

Can you see what to do to finish things off?
 
  • #6
On your last step I see you substituted x with y=3x+1 on the left-hand side, but not the right-hand side. In other words it should be 3-y (etc). Then you should have g(y) but you can set y=x to get g(x). To check your answer calculate f(g(h(x))) & see what you get.
 
  • #7
Nice I just got this. Thank you all for your help

g(x) = sqrt((10-x)/(7-x))

I spent this entire time trying to figure this out. I really couldn't find any similar examples on the internet.
 
  • #8
Poset said:

Homework Statement



Let f(g(h(x))) = 1/(2-x)

Find g(x) if:

f(x) = (x^2) - 1
h(x) = 3x+12. The attempt at a solution

This is what I have:

g(h(x))^2 -1 = 1/(2-x)
g(h(x) = sqrt((3-x)/(2-x))

I'm not sure how to get the h(x) out of this to leave me with just g(x). Please point me in the right direction.

Thank you!

Do it systematically: ##f(g(h(x)) = g(h(x))^2 - 1## and ##g(h(x)) = g(3x+1)##, so ##f(g(h(x)) = g(3x-1)^2 - 1##.
 
  • #9
PeroK said:
You've got:

##g(h(x)) = \sqrt{\frac{3-x}{2-x}}##

Why not take one more step?

##g(3x+1) = \sqrt{\frac{3-x}{2-x}}##

Can you see what to do to finish things off?
I like PeroK's reply here.

Two things to point out to Poset:

First of all; f(x) is not a 1 to 1 function.

So, when you find ##g(h(x)) = \sqrt{\frac{3-x}{2-x}}## , you should include a ± , so you can choose either sign in finding g(x). Thus the choice for g(x) is not unique.

##\displaystyle g(h(x)) = \pm\sqrt{\frac{3-x}{2-x}}##​

Second: The composite function, ##\ f\circ g\circ h\ ##, may have a more restricted domain than the implicit domain of ##\ \displaystyle k(x)=\frac{1}{2-x}\ ##.

Added in Edit: I've been working on this while the recent posts have come in.

Note: ##\displaystyle\ \frac{10-x}{7-x}=\frac{x-10}{x-7}##
 
Last edited:
  • #10
I thought this might be of interest.

Below is a graph of the composite function y = f(g(h(x))), (in black). The range is restricted to y ≥ -1, because the final function, f, has a range, [-1,∞) .

The missing piece of the graph for y =1/(2-x) is shown dotted in grey.

Triple composition.png

.
 
  • #11
SammyS said:
I thought this might be of interest.

Below is a graph of the composite function y = f(g(h(x))), (in black). The range is restricted to y ≥ -1, because the final function, f, has a range, [-1,∞) .

The missing piece of the graph for y =1/(2-x) is shown dotted in grey.

View attachment 80159
.

Just as a matter of interest: what package did you use to make the graph?
 
  • #12
Ray Vickson said:
Just as a matter of interest: what package did you use to make the graph?
Ray,

It's called Graph - appropriately enough.

Graph_prgm.PNG

It's not perfect, but usually gets the job done.
 
  • #13
SammyS said:
Ray,

It's called Graph - appropriately enough.

View attachment 80164
It's not perfect, but usually gets the job done.

Thank you for that.
 

Related to Longtime lurker here, composite function problem

1. What is a composite function?

A composite function is a mathematical function that is formed by combining two or more functions. The output of one function becomes the input of another function, creating a chain of functions that are performed one after the other.

2. How do you solve a composite function problem?

To solve a composite function problem, you must first identify the individual functions that make up the composite function. Then, plug in the input value into the first function and use the output as the input for the second function. Continue this process until you reach the final output value.

3. What is the purpose of using composite functions?

Composite functions are useful in simplifying complex mathematical equations and modeling real-world situations. They also allow us to break down a larger problem into smaller, more manageable parts.

4. Can a composite function have more than two functions?

Yes, a composite function can have any number of functions. As long as the output of one function is compatible with the input of the next function, you can continue to add more functions to create a composite function.

5. Are there any specific rules for composing functions?

Yes, there are a few rules to keep in mind when composing functions. The output of one function must be the same type of input that the next function can accept. Also, the order in which the functions are composed can affect the overall result, so it is important to pay attention to the order of operations.

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