Transformations and Function Compositions Checkpoint

Front

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)).

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.

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.

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.

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)).

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.

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).

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.

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.