Composite Function Homework: Is My Solution Correct?

In summary: Thanks for the help. You cleared it up for me.In summary, based on the given information, the correct way to solve this question is to use a guessing approach and then use the step above to find the correct answer.
  • #1
Precursor
222
0
Homework Statement
141sgif.jpg


The attempt at a solution

[tex]g(f(x)) = h(x)[/tex]
[tex]4f(x) + y = 4x - 1[/tex]
[tex]4x + 16 + y = 4x - 1[/tex]
[tex]y = -1 - 16[/tex]
[tex]y = -17[/tex]

so, [tex]g(x)= 4x + y = 4x - 17[/tex]

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.
 
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  • #2
Precursor said:
Homework Statement
141sgif.jpg


The attempt at a solution

[tex]g(f(x)) = h(x)[/tex]
==> 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.
Precursor said:
[tex]4f(x) + y = 4x - 1[/tex]
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.
Precursor said:
[tex]4x + 16 + y = 4x - 1[/tex]
[tex]y = -1 - 16[/tex]
[tex]y = -17[/tex]

so, [tex]g(x)= 4x + y = 4x - 17[/tex]

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.
 
  • #3
Thanks for the help. You cleared it up for me.
 
  • #4
Mark44 said:
==> 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.
 

1. How do I know if my solution for a composite function is correct?

In order to determine if your solution for a composite function is correct, you should first check if your final answer matches the given function. Then, you should substitute your solution into both functions and see if the resulting output is the same. If it is, then your solution is likely correct.

2. Can I use different methods to solve a composite function?

Yes, there are multiple methods that can be used to solve a composite function. Some common methods include using the associative property, the distributive property, or the composite function formula. It is important to choose a method that you are comfortable with and that will give you the most accurate solution.

3. What should I do if my solution for a composite function is incorrect?

If your solution for a composite function is incorrect, you should retrace your steps and double-check your calculations. It is also helpful to ask for assistance from a classmate or your teacher. You may have made a mistake in your calculations or used the wrong method to solve the function.

4. How can I check my work for a composite function?

One way to check your work for a composite function is to use a graphing calculator. You can enter both functions and compare the resulting graphs to see if they match. Another way is to use a table of values, where you can plug in different values and compare the outputs from both functions.

5. Are there any common mistakes that people make when solving a composite function?

Yes, some common mistakes when solving a composite function include forgetting to substitute the variable with the given value, using the wrong method to solve the function, and making errors in calculations. It is important to carefully follow the steps and check your work to avoid these mistakes.

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