Question

If \(f(x)=\sqrt{x}\) and \(g(x)=x^{3}-2,\) find the compositions \(f \circ g, g \circ f, f \circ f,\) and \(g \circ g\).

Short answer

Answer: \(f(g(x)) = \sqrt{x^3 - 2}\) 2. Given the functions \(f(x) = \sqrt{x}\) and \(g(x) = x^3 - 2\), what is the composite function \(g(f(x))\)? Answer: \(g(f(x)) = x^{\frac{3}{2}} - 2\) 3. Given the function \(f(x) = \sqrt{x}\), what is the composite function \(f(f(x))\)? Answer: \(f(f(x)) = \sqrt[4]{x}\) 4. Given the function \(g(x) = x^3 - 2\), what is the composite function \(g(g(x))\)? Answer: \(g(g(x)) = x^9 - 6x^6 + 12x^3 - 10\)

Step-by-step solution

Find \(f(g(x))\) (or \(f \circ g\))

To find \(f(g(x))\), substitute the expression for \(g(x)\) into the argument of \(f(x)\). This means replacing \(x\) in \(f(x)\) with the expression of \(g(x)\): \(f(g(x)) = f(x^3 - 2) = \sqrt{x^3 - 2}\)

Find \(g(f(x))\) (or \(g \circ f\))

To find \(g(f(x))\), substitute the expression for \(f(x)\) into the argument of \(g(x)\). This means replacing \(x\) in \(g(x)\) with the expression of \(f(x)\): \(g(f(x)) = g(\sqrt{x}) = (\sqrt{x})^3 - 2 = x^{\frac{3}{2}} - 2\)

Find \(f(f(x))\) (or \(f \circ f\))

To find \(f(f(x))\), substitute the expression for \(f(x)\) into the argument of itself: \(f(f(x)) = f(\sqrt{x}) = \sqrt{\sqrt{x}} = \sqrt[4]{x}\)

Find \(g(g(x))\) (or \(g \circ g\))

To find \(g(g(x))\), substitute the expression for \(g(x)\) into the argument of itself: \(g(g(x)) = g(x^3 - 2) = (x^3 - 2)^3 - 2 = x^9 - 6x^6 + 12x^3 - 10\) So, the compositions of the given functions are: 1. \(f \circ g\): \(f(g(x)) = \sqrt{x^3 - 2}\) 2. \(g \circ f\): \(g(f(x)) = x^{\frac{3}{2}} - 2\) 3. \(f \circ f\): \(f(f(x)) = \sqrt[4]{x}\) 4. \(g \circ g\): \(g(g(x)) = x^9 - 6x^6 + 12x^3 - 10\)

Key concepts

Composite Functions

In mathematics, composite functions represent the idea of combining two functions to create a new one. Think of it as a two-step process; whatever the first function does, the second function takes that result and does something further to it.

Creating a composite function involves taking the output from one function and using it as the input for another, symbolized as \(f \circ g\) which reads as 'f composed with g'. The only rule you must remember is that the order matters – performing \(f \circ g\) might yield a completely different result from \(g \circ f\), just as putting on socks before shoes leads to a different outcome than shoes before socks!

Calculus

Calculus is a branch of mathematics concerned with the study of change and motion. Through its two major subfields, differential and integral calculus, it provides tools for describing and analyzing the behavior of functions.

When dealing with composite functions in calculus, we often look into their derivatives and integrals, which require the application of the chain rule for differentiation or u-substitution for integration. These techniques allow us to handle complex functions made from simpler ones and are essential for solving real-world problems involving rates of change and areas under curves.

Function Operations

Just like numbers, functions can undergo operations. These include addition, subtraction, multiplication, division, and composition. During these operations, we combine functions in different ways to produce new functions.

For instance, when we talk about function composition in our exercise, we are in essence creating a new function by applying one function to the result of another. Unlike addition or multiplication, which can be done in any order (commutative), composition is not always commutative; the order in which functions are composed is critical.

Square Root Function

The square root function, denoted as \(f(x) = \sqrt{x}\), is a special type of function that answers the question: 'What number multiplied by itself would give us x?'. In our exercise, when we compose the square root function with another function, we are essentially finding the square root of whatever the other function outputs.

This function is the inverse of squaring a number and is crucial in solving equations where the variable is under a square root. Moreover, the graph of this function is a curve that increases slowly and never goes past the x-axis, reflecting the fact that square roots of positive numbers are always positive or zero.

Polynomial Functions

Polynomial functions are mathematical expressions involving a sum of powers of the variable, like \(g(x) = x^3 - 2\) in our exercise. They can be simple, with just one term, or very complex with many terms and different powers.

Polynomials are fundamental in calculus and algebra because they are simple enough to be easily understood and manipulated, yet versatile enough to model a vast range of phenomena. The operations with these functions follow algebraic rules and when composed, as seen in \(g \circ g\), can generate even higher degree polynomials with new coefficients, reflecting the compounded effect of applying the function to itself.