Podcast
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?
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.
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?
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?
Flashcards
f(g(x))
f(g(x))
This involves substituting the entire function g(x) into the variable x of the function f(x).
g(f(x))
g(f(x))
This involves substituting the entire function f(x) into the variable x of the function g(x).
Finding f(g(5))
Finding f(g(5))
To calculate f(g(5)), first evaluate g(5) using the function g(x)=x-2. Then plug the result into f(x)=4x+3.
Finding g(f(-6)).
Finding g(f(-6)).
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Finding f(f(7))
Finding f(f(7))
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Function Composition
Function Composition
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Composition (gof)(x)
Composition (gof)(x)
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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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