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Questions and Answers
What is the result of f(g(5)) if f(x)=4x+3 and g(x)=x-2?
What is the result of f(g(5)) if f(x)=4x+3 and g(x)=x-2?
15
What do you get when you calculate g(f(-6)) for f(x)=4x+3 and g(x)=x-2?
What do you get when you calculate g(f(-6)) for f(x)=4x+3 and g(x)=x-2?
-23
What is the result of f(f(7)) if f(x)=4x+3?
What is the result of f(f(7)) if f(x)=4x+3?
127
What is the composition g(f(x)) if f(x)=4x+3 and g(x)=x-2?
What is the composition g(f(x)) if f(x)=4x+3 and g(x)=x-2?
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Calculate (fog)(x) if f(x)=6x^2 and g(x)=14x+4.
Calculate (fog)(x) if f(x)=6x^2 and g(x)=14x+4.
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What is the result of (gof)(x) for f(x)=6x^2 and g(x)=14x+4?
What is the result of (gof)(x) for f(x)=6x^2 and g(x)=14x+4?
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Are the results of g(f(-6)) and f(g(5)) the same?
Are the results of g(f(-6)) and f(g(5)) the same?
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Study Notes
Composition of Functions
-
Function Definitions:
- ( f(x) = 4x + 3 )
- ( g(x) = x - 2 )
-
Example Calculation:
- For ( f(g(5)) ):
- Calculate ( g(5) = 5 - 2 = 3 )
- Then ( f(3) = 4(3) + 3 = 15 )
- For ( f(g(5)) ):
-
Example Calculation:
- For ( g(f(-6)) ):
- Find ( f(-6) = 4(-6) + 3 = -21 )
- Then ( g(-21) = -21 - 2 = -23 )
- For ( g(f(-6)) ):
-
Example Calculation:
- For ( f(f(7)) ):
- Calculate ( f(7) = 4(7) + 3 = 31 )
- Then ( f(31) = 4(31) + 3 = 127 )
- For ( f(f(7)) ):
-
Composite Function:
- For ( g(f(x)) ):
- Start with ( f(x) = 4x + 3 )
- Then calculate ( g(4x + 3) = (4x + 3) - 2 = 4x + 1 )
- For ( g(f(x)) ):
Alternative Functions
-
Using Different Functions:
- If ( f(x) = 6x^2 ) and ( g(x) = 14x + 4 ):
- Composition ( (f \circ g)(x) ):
- Calculate ( g(x) = 14x + 4 )
- Then ( f(g(x)) = f(14x + 4) = 6(14x + 4)^2 = 1176x^2 + 672x + 48 )
- Composition ( (f \circ g)(x) ):
- If ( f(x) = 6x^2 ) and ( g(x) = 14x + 4 ):
-
Other Composition:
- For ( (g \circ f)(x) ):
- Start with ( f(x) = 6x^2 )
- Then calculate ( g(f(x)) = g(6x^2) = 14(6x^2) + 4 = 84x^2 + 4 )
- For ( (g \circ f)(x) ):
Key Observations
-
Difference in Results:
- The results of ( (f \circ g)(x) ) and ( (g \circ f)(x) ) are not the same.
- This demonstrates that function composition is not commutative; the order matters.
- Understanding composition typically requires familiarity with the specific functions involved.
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Description
Test your understanding of function composition with this quiz on f(x) and g(x). The questions involve evaluating combined functions and their impacts. Perfect for students looking to solidify their knowledge of function operations.
