Problem 36 - College Algebra

Question

Find: a. \((f \circ g)(x)\) b. the domain of \(f \circ g\) $$f(x)=x^{2}+1, g(x)=\sqrt{2-x}$$

Step-by-Step Solution

Answer

a. \((f \circ g)(x)\) is \(3-x\).\n b. The domain of \(f \circ g\) is \(x \leq 2\) or \([-∞, 2]\).

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

To find \((f \circ g)(x)\), you need to put \(g(x)\) inside \(f(x)\). Therefore, \[f(g(x)) = f(\sqrt{2-x}) = {(\sqrt{2-x})}^2+1 = 2-x+1 = 3-x\] So, \((f \circ g)(x)\) equals \(3-x\).

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

The domain of the composite function should satisfy the conditions of both \(f(x)\) and \(g(x)\). Given that \(f(x)\) is a polynomial function, its domain is all real numbers. However, \(g(x)\) is a square root function, it only accepts real numbers and its argument must be greater than or equal to zero. So, \(2-x \geq 0 \). Solving this inequality for \(x\), \(x \leq 2\). Hence, the domain of \(f \circ g\) is \(x \leq 2\), or we can express it in interval notation as \([-∞, 2]\).

Key Concepts

Domain of a Function

The domain of a function refers to all the possible input values (or "x" values) that make the function valid and produce a real output. To find the domain, you need to consider any mathematical operations in the function's formula that might restrict certain values.
For instance, when dealing with fractions, you'd want to avoid a zero in the denominator. Meanwhile, with square root functions, the expression inside the square root must be non-negative.
To determine the domain of a composite function, you need to check the domains of both individual functions involved before composition.
This consideration ensures that all operations conducted on initial input values are valid for every part of the function.

Function Composition

Function composition involves creating a new function by applying one function to the results of another. It's like putting one function inside another, which is denoted as \( (f \circ g)(x) = f(g(x)) \).
To perform function composition:

• Start by evaluating the inner function, \(g(x)\), using your input value \(x\).
• Then, use this resulting value from \(g(x)\) as the input for the outer function \(f(x)\).
• Write the final expression of \(f(g(x))\) by substituting \(g(x)\) into the formula of \(f(x)\).

In our example, \(g(x)\) was substituted into \(f(x)\) to get \((f \circ g)(x) = 3-x\). Remember, function composition demands understanding both functions and how they interact when combined.

Polynomial Functions

Polynomial functions are mathematical expressions involving terms with variables raised to whole number exponents and having coefficients. For instance, \(f(x) = x^2 + 1\) is a typical polynomial.
Some key characteristics of polynomial functions:

• Their graph is continuous and infinitely smooth (no sharp corners or breaks).
• They can have a domain of all real numbers, meaning any real number can be used as input.
• The degree of the polynomial (highest exponent value) determines its overall shape and behavior.

These properties make polynomials versatile and integral in various mathematical computations, ensuring you can use them without restrictions on many sets of input values. Whether they're linear or quadratic like in our example, they play a crucial role in function analysis.

Square Root Functions

Square root functions involve taking the square root of an expression. Such functions have a specific domain restriction because you can only take the square root of non-negative numbers in the real number system. In our example, \(g(x) = \sqrt{2-x}\), the argument \(2-x\) must be greater than or equal to zero.
To find the domain of a square root function:

• Set the expression inside the square root (the radicand) \(\geq 0\).
• Solve this inequality to determine the acceptable values for \(x\).

In our exercise, we solved \(2-x \geq 0\), resulting in \(x \leq 2\). Therefore, the domain of \(g(x)\) is \([-∞, 2]\). Square root functions require careful domain consideration to maintain real-number results.