Transformations and Function Compositions Checkpoint

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What is a function composition?

Back

A function composition is the application of one function to the results of another function. It is denoted as (f ∘ g)(x) = f(g(x)).

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CCSS.HSF-BF.A.1C

Front

How do you find the composition of two functions f and g?

Back

To find the composition f ∘ g, substitute g(x) into f. For example, if f(x) = 2x and g(x) = x + 3, then f ∘ g(x) = f(g(x)) = f(x + 3) = 2(x + 3) = 2x + 6.

Tags

CCSS.HSF-BF.A.1C

Front

What is the difference between f(g(x)) and g(f(x))?

Back

f(g(x)) applies g first and then f to the result, while g(f(x)) applies f first and then g. The order of operations matters and can lead to different results.

Tags

CCSS.HSF-BF.A.1C

Front

What does it mean for a function to be linear?

Back

A linear function is a function that can be graphically represented as a straight line. It has the form f(x) = mx + b, where m is the slope and b is the y-intercept.

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CCSS.8.F.B.4

CCSS.HSF.LE.A.2

Front

How do you determine the composition function f ∘ g from their graphs?

Back

To determine f ∘ g from the graphs, find the output of g for a given input x, then use that output as the input for f. The resulting value is f(g(x)).

Tags

CCSS.HSF-BF.A.1C

Front

What is the significance of the order of operations in function composition?

Back

The order of operations in function composition affects the outcome. For example, applying g first may yield a different result than applying f first.

Tags

CCSS.HSF-BF.A.1C

Front

What is the graphical interpretation of shifting a function?

Back

Shifting a function vertically or horizontally involves adding or subtracting a constant to/from the function's input or output, respectively. For example, f(x - 4) shifts the graph of f left by 4 units.

Front

How do you evaluate g(f(2)) if f(2) is known?

Back

To evaluate g(f(2)), first find the value of f(2), then substitute that value into g. For example, if f(2) = 3, then g(f(2)) = g(3).

Tags

CCSS.HSF-BF.A.1C

Front

What is the effect of applying a discount coupon as a function?

Back

Applying a discount coupon can be modeled as a function where the input is the original price and the output is the discounted price. For example, f(x) = x - 10 for a $10 off coupon.

Tags

CCSS.8.F.B.4

CCSS.HSF.LE.A.2

Front

What is the relationship between the graphs of f(x) and g(x) if g(x) = f(x + c)?

Back

If g(x) = f(x + c), the graph of g is a horizontal shift of the graph of f. If c is positive, g shifts left; if c is negative, g shifts right.