Question

Composite functions and notation Let $f(x)=x^{2}-4$ $g(x)=x^{3},$ and $F(x)=1 /(x-3) .$ Simplify or evaluate the following expressions. $$F(g(y))$$

   Composite functions and notation Let $f(x)=x^{2}-4$ $g(x)=x^{3},$ and $F(x)=1 /(x-3) .$ Simplify or evaluate the following expressions.
$$F(g(y))$$

Calculus for Scientists and Engineers: Early Transcendental

William Briggs, Lyle… 1st Edition

Chapter 1, Problem 29

Instant Answer

Solved by Expert Charles Carter

05/28/2020

Step 1

It wants us to find the composite function $F(g(y))$. This means we need to substitute $g(y)$ into the function $F(x)$. Show more…

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Composite functions and notation Let $f(x)=x^{2}-4$ $g(x)=x^{3},$ and $F(x)=1 /(x-3) .$ Simplify or evaluate the following expressions. $$F(g(y))$$

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Key Concepts

Substitution in Function Composition

Substitution is a central idea in evaluating composite functions. It requires replacing the variable in the outer function with the complete expression of the inner function. This method enables the simplification or evaluation of the composite expression by 'substituting' one function into another.

Composite Functions

Composite functions involve applying one function to the results of another function. This means that the output of one function becomes the input of another, represented as (F ? g)(y) = F(g(y)). This key concept is fundamental in understanding how functions interact when they are combined in a composition.

Function Notation

Function notation provides a way to succinctly denote the relationship between inputs and outputs. It allows us to express operations like composition, where the expression F(g(y)) indicates that the function g is applied to y first, and then function F is applied to the result.

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