key term - F(g(x))

Definition

The term f(g(x)) represents the composition of two functions, where the inner function g(x) is evaluated first, and then the result is used as the input for the outer function f(x). This allows for the combination of two separate functions into a single, more complex function.

5 Must Know Facts For Your Next Test

1. The order of the functions in f(g(x)) is important, as the inner function g(x) is evaluated first, and its result is then used as the input for the outer function f(x).
2. The domain of the composite function f(g(x)) is the set of all x values in the domain of g(x) for which g(x) is in the domain of f(x).
3. The range of the composite function f(g(x)) is the set of all possible output values of f(g(x)).
4. Composing functions can be a useful technique for solving complex problems by breaking them down into simpler steps.
5. Inverse functions can be used to undo the effect of function composition, allowing you to solve for the original input value.

Review Questions

• Explain the process of evaluating the expression f(g(x)), step-by-step.
  • To evaluate the expression f(g(x)), you first evaluate the inner function g(x) to obtain a value. Then, you use that value as the input for the outer function f(x) and evaluate it. The result of this two-step process is the final value of the composite function f(g(x)). For example, if g(x) = x^2 and f(x) = 3x + 1, then to evaluate f(g(x)) for x = 2, you would first evaluate g(2) = 2^2 = 4, and then evaluate f(4) = 3(4) + 1 = 13. Therefore, f(g(2)) = 13.
• Describe how the domains and ranges of the individual functions f(x) and g(x) impact the domain and range of the composite function f(g(x)).
  • The domain of the composite function f(g(x)) is the set of all x values in the domain of g(x) for which g(x) is in the domain of f(x). In other words, the composite function f(g(x)) is only defined for the values of x where both the inner function g(x) and the outer function f(x) are defined. The range of the composite function f(g(x)) is the set of all possible output values of f(g(x)), which is determined by the ranges of the individual functions f(x) and g(x). Restrictions on the domains and ranges of the component functions can limit the domain and range of the composite function.
• Explain how the concept of function composition can be used to solve complex problems by breaking them down into simpler steps.
  • Function composition allows you to combine multiple functions to create a more complex function that can be used to solve intricate problems. By breaking down a complex problem into a series of simpler steps, each represented by a function, you can use function composition to solve the overall problem. For example, if you need to find the area of a circle with a radius that is dependent on the value of another variable, you can use function composition to first find the radius as a function of the other variable, and then use that result as the input to the formula for the area of a circle. This stepwise approach, facilitated by function composition, can be a powerful problem-solving technique.

Related terms

Function Composition:

The process of combining two or more functions to create a new function, where the output of one function becomes the input of the next function.

Domain and Range:

The set of input values (domain) and output values (range) for a function, which are important considerations when composing functions.

Inverse Functions:

Functions that undo the effect of another function, which can be useful when working with function composition.