Composition of functions

Front

What is a function composition?

Back

Function composition is the process of applying one function to the results of another function. If f(x) and g(x) are two functions, then the composition of f and g is denoted as (f ° g)(x) = f(g(x)).

Front

How do you find (f ° g)(x) if f(x) = 3x - 1 and g(x) = x^2 + 2?

Back

To find (f ° g)(x), substitute g(x) into f: (f ° g)(x) = f(g(x)) = f(x^2 + 2) = 3(x^2 + 2) - 1 = 3x^2 + 6 - 1 = 3x^2 + 5.

Front

What is the result of f(g(x)) if f(x) = 2x + 3 and g(x) = -3x - 4?

Back

To find f(g(x)), substitute g(x) into f: f(g(x)) = f(-3x - 4) = 2(-3x - 4) + 3 = -6x - 8 + 3 = -6x - 5.

Front

How do you calculate (g - h)(x) if g(x) = 2x^2 - 5x + 1 and h(x) = -6x^2 + 5?

Back

To find (g - h)(x), subtract h from g: (g - h)(x) = g(x) - h(x) = (2x^2 - 5x + 1) - (-6x^2 + 5) = 2x^2 - 5x + 1 + 6x^2 - 5 = 8x^2 - 5x - 4.

Front

What is the value of f(g(8)) if f(x) = 2x + 3 and g(x) = -3x - 4?

Back

First, calculate g(8): g(8) = -3(8) - 4 = -24 - 4 = -28. Then, find f(g(8)): f(-28) = 2(-28) + 3 = -56 + 3 = -53.

Front

What does it mean for two functions to be inverses?

Back

Two functions f and g are inverses if f(g(x)) = x and g(f(x)) = x for all x in the domain of g and f, respectively.

Front

What is the domain of a function composition?

Back

The domain of the composition (f ° g)(x) is the set of all x in the domain of g such that g(x) is in the domain of f.

Front

How do you evaluate (f ° g)(x) for f(x) = x^2 and g(x) = x + 1?

Back

To evaluate (f ° g)(x), substitute g(x) into f: (f ° g)(x) = f(g(x)) = f(x + 1) = (x + 1)^2 = x^2 + 2x + 1.

Front

What is the graphical representation of function composition?

Back

Graphically, the composition of functions can be represented by plotting the output of g(x) on the x-axis and then plotting f(g(x)) on the y-axis.

Front

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

Back

(f ° g)(x) means applying g first and then f, while (g ° f)(x) means applying f first and then g. They are generally not equal.