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

Question: Find the compositions of functions \(f \circ g\), \(g \circ f\), \(f \circ f\), and \(g \circ g\) where \(f(x) = \sqrt{x}\) and \(g(x)=x^3-2\). Answer: 1. \(f \circ g(x) = \sqrt{x^3 - 2}\) 2. \(g \circ f(x) = x\sqrt{x} - 2\) 3. \(f \circ f(x) = \sqrt[4]{x}\) 4. \(g \circ g(x) = (x^3 - 2)^3 - 2\)

Step-by-step solution

Find \(f \circ g\)

To find \(f \circ g(x)\), we will plug \(g(x)\) into \(f(x)\): $$f(g(x)) = f(x^3 - 2)$$ Now, we will substitute the right side of the definition of \(f(x)\): $$f(g(x)) = \sqrt{x^3 - 2}$$

Find \(g \circ f\)

To find \(g \circ f(x)\), we will plug \(f(x)\) into \(g(x)\): $$g(f(x)) = g(\sqrt{x})$$ Now, substitute the right side of the definition of \(g(x)\): $$g(f(x)) = (\sqrt{x})^3 - 2$$ Simplify the expression: $$g(f(x)) = x\sqrt{x} - 2$$

Find \(f \circ f\)

To find \(f \circ f(x)\), we will plug \(f(x)\) into \(f(x)\): $$f(f(x)) = f(\sqrt{x})$$ Now, substitute the right side of the definition of \(f(x)\): $$f(f(x)) = \sqrt{\sqrt{x}}$$ Simplify the expression: $$f(f(x)) = \sqrt[4]{x}$$

Find \(g \circ g\)

To find \(g \circ g(x)\), we will plug \(g(x)\) into \(g(x)\): $$g(g(x)) = g(x^3 - 2)$$ Now, substitute the right side of the definition of \(g(x)\): $$g(g(x)) = (x^3 - 2)^3 - 2$$ The compositions we have found are: 1. \(f \circ g(x) = \sqrt{x^3 - 2}\) 2. \(g \circ f(x) = x\sqrt{x} - 2\) 3. \(f \circ f(x) = \sqrt[4]{x}\) 4. \(g \circ g(x) = (x^3 - 2)^3 - 2\)

Key concepts

Square Root Function

The square root function is a mathematical function represented by the symbol \(\sqrt{x}\). It is the inverse operation of squaring a number. The square root of a number \(x\) is a value that, when multiplied by itself, gives \(x\). In function notation, the square root function \(f(x) = \sqrt{x}\) takes a non-negative number and returns its square root. This function is only defined for non-negative values of \(x\), as the square root of a negative number is not a real number.

• The domain of \(f(x) = \sqrt{x}\) is \(x \geq 0\).
• The range is \(f(x) \geq 0\).

For example, \(f(4) = \sqrt{4} = 2\), and \(f(9) = \sqrt{9} = 3\).
When composing the square root function with another function, such as in the expression \(f(g(x)) = \sqrt{g(x)}\), we substitute \(g(x)\) into the square root function. In our exercise, \(f(g(x)) = \sqrt{x^3 - 2}\), demonstrating how the square root function works with compositions.

Cubic Function

The cubic function is a type of polynomial function with the form \(g(x) = x^3 + a\) where 'a' is a constant. A simple example of a cubic function is \(g(x) = x^3\), and it is characterized by its degree, which is 3. The graph of a cubic function is typically "S" shaped and it spans from negative infinity to positive infinity, meaning its domain and range are all real numbers.

• The general form is \(y = ax^3 + bx^2 + cx + d\), where \(a eq 0\).
• It can have up to three real roots, also known as x-intercepts.
• The behavior at infinity is that as \(x\) goes to infinity, \(y\) goes to infinity or negative infinity, depending on the leading coefficient.

In the context of function composition, such as \(g(f(x))\), you plug the result of \(f(x)\) into \(g(x)\). For example, if \(f(x) = \sqrt{x}\), then \(g(f(x)) = (\sqrt{x})^3 - 2 = x\sqrt{x} - 2\), as simplified in our exercise.

Function Simplification

Function simplification involves expressing a function in its simplest form, making it easier to handle or interpret. Simplification can be done through various algebraic manipulations, such as factoring, combining like terms, or canceling terms.

• This often involves recognizing patterns like perfect squares or cubes.
• The goal is to make functions easier to compute or integrate into larger expressions.

In the exercise, simplification was seen in steps such as converting \((\sqrt{x})^3 = x\sqrt{x}\), which removes the unnecessary parenthesis and combines the terms effectively. Another example is the simplification of \(f(f(x)) = \sqrt{\sqrt{x}}\) into \(\sqrt[4]{x}\), showing the power of functions and composition to boil down complex expressions to a more manageable form. Simplification can lead to greater insights into the behavior and properties of functions, particularly when they are combined or nested as in composition.