Question

If \(f(x)=2 x^{3}-3 x^{2}+4 x-1\) and \(g(x)=2,\) find \((f \circ g)(x)\) and \((g \circ f)(x)\)

Short answer

(f \circ g)(x) = 11, (g \circ f)(x) = 2

Step-by-step solution

Understand the Function Composition

Function composition, denoted as \( (f \circ g)(x) \), means that each input to f is the output of g. Similarly, \( (g \circ f)(x) \) means each input to g is the output of f.

Determine \( g(x) \)

Given \( g(x) = 2 \), this function value is constant for any input. Therefore, \( g(x) = 2 \) regardless of the value of \( x \).

Calculate \( (f \circ g)(x) \)

Plug \( g(x) \) into function f. Since \( g(x) = 2 \), substitute \( x = 2 \) into \( f(x) \): \[ f(x) = 2x^3 - 3x^2 + 4x - 1 \] \[ f(2) = 2(2^3) - 3(2^2) + 4(2) - 1 \] Simplify: \[ f(2) = 2 \times 8 - 3 \times 4 + 8 - 1 = 16 - 12 + 8 - 1 = 11 \] Therefore, \( (f \circ g)(x) = 11 \).

Calculate \( (g \circ f)(x) \)

Plug function \( f(x) \) into \( g \). Since \( g(x) = 2 \) is constant, \( g(f(x)) = 2 \) for any \( x \). Thus, \( g(f(x)) = 2 \).

Key concepts

Composite Functions

Composite functions are a fascinating concept in mathematics that allow you to combine two functions into a new one. Imagine you have two functions, say, \(f(x)\) and \(g(x)\). When you create a composite function like \((f \, \circ \, g)(x)\), you are actually plugging the output of \(g(x)\) into \(f(x)\). Essentially, you perform \(g(x)\) first and then use its result as the input for \(f(x)\).

For example, if \(g(x) = 2\) and \(f(x) = 2x^3 - 3x^2 + 4x - 1\), to find \((f \, \circ \, g)(x)\), you would substitute \(g(x)\) into \(f(x)\):

\[ \begin{align*} (f \, \circ \, g)(x) &= f(g(x)) \ (f(g(x)) &= f(2) \ &= 2(2^3) - 3(2^2) + 4(2) - 1 \ &= 11 \end{align*} \]

So, \((f \, \circ \, g)(x)\) results in a constant value of \(11\) regardless of your input \(x\). In summary, composite functions provide a powerful way to link operations together, making complex calculations simpler.

Polynomial Functions

Polynomial functions are like the building blocks of many mathematical expressions you encounter. They consist of terms that are sums of constants multiplied by variables raised to whole number powers.

For instance, consider \(f(x) = 2x^3 - 3x^2 + 4x - 1\). This is a polynomial function of degree 3 because the highest power of \(x\) is 3. Each term (like \(2x^3\) or \(-3x^2\)) contributes to the overall shape and behavior of the polynomial.

When working with polynomial functions, we often need to perform operations like addition, subtraction, and composition. For example, to find \((f \, \circ \, g)(x)\), if \(g(x) = 2\), we substitute \(2\) into \(f(x)\) and simplify:

\[ \begin{align*} f(2) &= 2(2^3) - 3(2^2) + 4(2) - 1 \ &= 16 - 12 + 8 - 1 \ &= 11 \end{align*} \]

Polynomial functions are versatile and can be used to approximate more complex functions, model real-world scenarios, and much more.

Constant Function

A constant function is one of the simplest types of functions you'll encounter, yet it holds significant importance in mathematics. A constant function outputs the same value regardless of the input. So if \(g(x) = 2\), it means \(g\) will always return 2, no matter what \(x\) is.

When dealing with composition, constant functions simplify the process greatly. For instance, if we need to find \( (g \, \circ \, f)(x)\) where \(g(x) = 2\) and \(f(x) = 2x^3 - 3x^2 + 4x - 1\):

\[\begin{align*} g(f(x)) &= g(anything) \ &=2 \end{align*}\]

In other words, \( g(f(x)) \) equals 2 regardless of what \(f(x)\) outputs. This illustrates the power and simplicity of constant functions in composite operations.