Problem 19

Question

Find $(f \circ g)(x)$ and $(g \circ f)(x)$. $$ f(x)=x^{3}+x-2 ; g(x)=-2 x $$

Step-by-Step Solution

Verified

Answer

$(f \circ g)(x) = -8x^3 - 2x - 2$, $(g \circ f)(x) = -2x^3 - 2x + 4$.

1Step 1: Understand Composition of Functions

The composition of two functions, say $(f \circ g)(x)$, means you have to substitute the function $g(x)$ into the function $f(x)$. Similarly, the composition $(g \circ f)(x)$ involves substituting $f(x)$ into $g(x)$.

2Step 2: Substitute g(x) into f(x)

Substitute $g(x) = -2x$ into $f(x) = x^3 + x - 2$. So $(f \circ g)(x) = f(g(x)) = f(-2x)$. Calculate:\[(f \circ g)(x) = (-2x)^3 + (-2x) - 2\]This simplifies to:\[(f \circ g)(x) = -8x^3 - 2x - 2\]

3Step 3: Substitute f(x) into g(x)

Substitute $f(x) = x^3 + x - 2$ into $g(x) = -2x$. Thus, $(g \circ f)(x) = g(f(x)) = g(x^3 + x - 2)$. Calculate:\[(g \circ f)(x) = -2(x^3 + x - 2)\]This simplifies to:\[(g \circ f)(x) = -2x^3 - 2x + 4\]

Key Concepts

Composed FunctionsSubstitution in FunctionsPolynomial Functions

Composed Functions

When dealing with composed functions, the main idea is to "stack" two or more functions together. Think of it as letting one function transform the input, and then passing this output into another function. In mathematical notation, a composition is written as $f \circ g$, which means $f(g(x))$. In this scenario:

Function $g(x)$ is executed first.
The result of $g(x)$ becomes the input for the function $f(x)$.

For the given exercise, you are to find $f(g(x))$ and $g(f(x))$. The challenge with composed functions is keeping track of which function acts first, but by following the order presented by the composition symbol, you can manage the task effectively.

Substitution in Functions

Substitution is a core skill when dealing with composed functions. In general, to substitute one function into another means to replace the variable in one function with another function's expression. For example, substituting $g(x)=-2x$ into $f(x)=x^3+x-2$ creates a new function where every instance of $x$ in $f(x)$ is replaced by $-2x$.

For $f \circ g$, substitute $g(x)$ into $f(x)$.
Results in $f(-2x)=(-2x)^3 + (-2x) - 2$.
Each step should be solved systematically, handle powers and sign changes carefully.

Similarly, substituting $f(x)$ into $g(x)$ involves replacing $x$ in $g(x)=-2x$ by $x^3+x-2$, simplifying the expression step-by-step:

For $g \circ f$, substitute $f(x)$ into $g(x)$.
Results in $-2(x^3 + x - 2) = -2x^3 - 2x + 4$.

Polynomial Functions

A polynomial function is a sum of terms, each consisting of a variable raised to a whole number power, multiplied by a coefficient. For example, $f(x)=x^3+x-2$ is a polynomial of degree 3, since the highest power of $x$ is 3. Such functions are significant in composition because they may increase or reduce in complexity depending on the function they are composed with.

The term $x^3$ means it has a cubic component.
Polynomial functions are closed under addition, subtraction, and multiplication.
This makes them predictable when combining functions or performing substitutions.

Understanding how each term affects the outcome after substitution is crucial. For instance, when multiplying or expanding terms in compositions, keeping track of coefficients and powers ensures accurate results.

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