For each function below: i) Determine possible functions f and g so that y = f(g(x)). ii) Determine possible functions f, g, and h so that y = f(g(h(x))). Create composite function... For each function below: i) Determine possible functions f and g so that y = f(g(x)). ii) Determine possible functions f, g, and h so that y = f(g(h(x))). Create composite functions using either or both functions in each pair of functions below. In each case, how many different composite functions could you create? Justify your answer.
Understand the Problem
The question involves determining possible functions f and g for given equations and creating composite functions based on those functions. It asks for an analysis of how to achieve certain expressions through function composition.
Answer
a) For parts a) and b): \( y = f(g(x)) = x^2 - 6x + 5 \) and \( y = f(g(x)) = -3x^2 - 30x - 40 \). b) 2 possible composite functions for each pair: \( f(g(x)) \) and \( g(f(x)) \).
Answer for screen readers
a) Possible functions:
- For part (a): ( y = f(g(x)) = x^2 - 6x + 5 )
- For part (b): ( y = f(g(x)) = -3x^2 - 30x - 40 )
- For part (c): ( y = f(g(x)) = \sqrt{x^2 - 2} + 3 )
- For part (d): ( y = f(g(x)) = \sqrt{x^2} + 4x + 3 )
b) Composite functions:
- For part (a): 2 composites possible
- For part (b): 2 composites possible
Steps to Solve
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Identifying Functions for Part (a)
For the function ( y = x^2 - 6x + 5 ), we can express it as a composite function ( y = f(g(x)) ).
A possible choice is:
- Let ( g(x) = x )
- Then ( f(x) = x^2 - 6x + 5 )
Therefore, ( y = f(g(x)) = f(x) = x^2 - 6x + 5 ).
-
Identifying Functions for Part (b)
For the function ( y = -3x^2 - 30x - 40 ), we can choose:
- Let ( g(x) = x )
- Then ( f(x) = -3x^2 - 30x - 40 )
So, ( y = f(g(x)) = f(x) = -3x^2 - 30x - 40 ).
-
Identifying Functions for Part (c)
For ( y = \sqrt{x^2 - 2} + 3 ):
- Let ( g(x) = x^2 - 2 )
- Then ( f(x) = \sqrt{x} + 3 )
Therefore, ( y = f(g(x)) = f(x^2 - 2) = \sqrt{x^2 - 2} + 3 ).
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Identifying Functions for Part (d)
For ( y = \sqrt{x^2} + 4x + 3 ):
- Let ( g(x) = x^2 )
- Then ( f(x) = \sqrt{x} + 4x + 3 )
Thus, ( y = f(g(x)) = f(x^2) = \sqrt{x^2} + 4x + 3 ).
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Creating Composite Functions for Part (a) and (b)
a) For ( f(x) = |x| ) and ( g(x) = \frac{1}{x} ):
- You can create ( f(g(x)) = \left|\frac{1}{x}\right| ) and ( g(f(x)) = \frac{|x|}{1} ).
b) For ( f(x) = \sqrt{x} ) and ( g(x) = |x| ):
- You can create ( f(g(x)) = \sqrt{|x|} ) and ( g(f(x)) = |\sqrt{x}| ).
a) Possible functions:
- For part (a): ( y = f(g(x)) = x^2 - 6x + 5 )
- For part (b): ( y = f(g(x)) = -3x^2 - 30x - 40 )
- For part (c): ( y = f(g(x)) = \sqrt{x^2 - 2} + 3 )
- For part (d): ( y = f(g(x)) = \sqrt{x^2} + 4x + 3 )
b) Composite functions:
- For part (a): 2 composites possible
- For part (b): 2 composites possible
More Information
Function composition is a key concept in mathematics where the output of one function becomes the input of another. This approach is particularly useful in complex equations, allowing for simplification and clearer understanding of relationships between functions.
Tips
- Confusing the order of functions. Remember that ( f(g(x)) ) means you apply ( g ) first, then ( f ).
- Not checking if the resulting composite function matches the original function given.
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