Problem 38 - College Algebra

Question

Find: a. \((f \circ g)(x)\) b. the domain of \(f \circ g\) $$f(x)=9-x^{2}, g(x)=\sqrt{x^{2}-9}$$

Step-by-Step Solution

Answer

The function \(f \circ g(x) = 18 - x^{2}\), and its domain is \([- \infty, -3] \cup [3, \infty]\).

Step 1: Find \(f \circ g\)

To find the function \(f \circ g\), replace every instance of \(x\) in \(f(x)\) with \(g(x)\). That gives us \(f(g(x)) = 9 - (\sqrt{x^{2}-9})^2\). Simplifying this gives us \(9 - (x^2 - 9) = -x^{2} + 18 = 18 - x^{2}\). So \(f \circ g = 18 - x^{2}\).

Step 2: Find the domain of \(f \circ g\)

The domain of the composition \(f \circ g\) is the set of all \(x\) for which \(g(x)\) is defined, and \(f(g(x))\) is also defined. Looking at the function \(g(x)=\sqrt{x^{2}-9}\), it is defined for \(x \geq 3\) and \(x \leq -3\). Using these values in \(f(g(x)) = 18 - x^{2}\), it is found that \(f(g(x))\) is defined for all these \(x\) values. So, the domain of \(f \circ g\) is \([- \infty, -3] \cup [3, \infty]\).

Step 3: Form the final answer

Combine the results of step 1 and step 2 to form the final answer.

Key Concepts

Domain of a Function

A function's domain is the set of all possible input values (usually "x" values) that allow the function to work without any undefined behavior.

• Understanding the domain is crucial because it tells us where the function exists and is ready to operate.
• For example, the function \(g(x) = \sqrt{x^2 - 9}\) requires a domain where the expression inside the square root is non-negative.

This means we need \(x^2 - 9 \geq 0\), which refines to \(x \geq 3\) or \(x \leq -3\).
These constraints ensure the function does not attempt to find the square root of a negative number, which is undefined in the real number system.
In our exercise, the domain of the composition \(f \circ g\) inherits the constraints of \(g(x)\) and is further limited by where \(f(g(x))\) makes sense. This results in a domain of \([-\infty, -3] \cup [3, \infty]\).

Function Notation

Function notation is a systematic way of expressing functions and their operations.

• When we write \(f(x)\), \(f\) names the function and \(x\) specifies the variable being used.
• Function notation clearly shows input-output relationships, making it easy to plug in values and perform operations.

In the realm of compositions, \((f \circ g)(x)\) represents applying \(g(x)\) first and then \(f\) to the result of \(g(x)\).
This notation helps in understanding which function's output acts as the input for another and is essential for tracking domain considerations and transformations.

Square Root Function

A square root function extracts the non-negative square root of a number or expression.

• The general form is \(\sqrt{x}\), which means finding a number \(y\) such that \(y^2 = x\).
• This function is only defined for inputs where \(x \geq 0\), ensuring the result stays within real numbers.

With \(g(x) = \sqrt{x^2 - 9}\), the challenge is to ensure \(x^2 - 9 \geq 0\).
This condition confirms the square root expression will not receive a negative number, keeping calculations valid.
Square root functions are essential in compositions, as they can restrict the domains based on the need for non-negative inputs.

Polynomial Function

Polynomial functions are algebraic expressions involving sums of powers of a variable.

• A typical polynomial looks like \(a_nx^n + a_{n-1}x^{n-1} + ... + a_1x + a_0\), where the coefficients \(a\) are real numbers and \(n\) is a non-negative integer.
• They are incredibly flexible, with applications in modeling, physics, and engineering.

In our example, \(f(x) = 9 - x^2\) is a simple polynomial where terms involve the square of \(x\) and a constant.
When dealing with polynomial functions in compositions like \(f(g(x)) = 18 - x^2\), it’s about substituting and simplifying expressions meticulously.
Understanding polynomials' roles adds depth to recognizing how compositions transform input variables into other functional forms.