Problem 60 - College Algebra with Modeling and Visualization

Question

Exercises \(57-72:\) Use the given \(f(x)\) and \(g(x)\) to find each of the following. Identify its domain. $$ \begin{array}{llll} \text { (a) }(f \circ g)(x) & \text { (b) }(g \circ f)(x) & \text { (c) }(f \circ f)(x) \end{array} $$ $$ f(x)=x^{2}, \quad g(x)=\sqrt{1-x} $$

Step-by-Step Solution

Answer

(a) 1-x, x ≤ 1; (b) √(1-x²), -1 ≤ x ≤ 1; (c) x⁴, all real x.

Step 1: Identifying (f \, \circ \, g)(x)

First, identify the function \(f(g(x)) = f(\sqrt{1-x})\). Since \(f(x) = x^2\), we substitute \(\sqrt{1-x}\) into \(f(x)\), giving us \(f(\sqrt{1-x}) = (\sqrt{1-x})^2 = 1-x\).

Step 2: Finding Domain of (f \, \circ \, g)(x)

The function \(g(x) = \sqrt{1-x}\) requires that \(1-x \geq 0\), meaning \(x \leq 1\). Thus, the domain of \(f \circ g\) is \[ x \in (-\infty, 1] \].

Step 3: Identifying (g \, \circ \, f)(x)

Next, find \(g(f(x)) = g(x^2)\). We substitute \(x^2\) into \(g(x) = \sqrt{1-x}\), giving us \(g(x^2) = \sqrt{1-x^2}\).

Step 4: Finding Domain of (g \, \circ \, f)(x)

The expression under the square root \(1-x^2\) must be non-negative: \(1-x^2 \geq 0\). This simplifies to \(-1 \leq x \leq 1\). Thus, the domain of \(g \circ f\) is \[ x \in [-1, 1] \].

Step 5: Identifying (f \, \circ \, f)(x)

Find \(f(f(x)) = f(x^2)\). Substitute \(x^2\) into \(f(x) = x^2\), giving us \(f(x^2) = (x^2)^2 = x^4\).

Step 6: Finding Domain of (f \, \circ \, f)(x)

Since \(f(x) = x^2\) defines valid outputs for all real numbers, the domain of \(f \circ f\) is all real numbers, \[ x \in (-\infty, \infty) \].

Key Concepts

Domain of a Function

The domain of a function represents the set of input values for which the function is defined. In other words, these are the values of the variable (usually denoted as "x") for which the function produces a valid output, or "y".
Understanding the domain is a crucial part of analyzing any mathematical function.

• For a basic function like a line, quadratic, or polynomial, the domain is often all real numbers, denoted as \( x \in (-\infty, \infty) \).
• However, certain functions have restrictions. For example, a function involving a fraction might exclude values that make the denominator zero.

When identifying domains, you may need to consider multiple aspects of the function:

• Absolute values require inputs that ensure non-negative results within a square root.
• Logarithmic functions necessitate positive inputs only, and no zeroes or negative values.

The task is to find these rules and apply them to determine which values x can assume.
This can involve setting inequalities and solving for variable ranges, as shown in our solution, such as the domains for \(f(g(x))\) and \(g(f(x))\).

Square Root Function

The square root function is a key mathematical concept defined by \( g(x) = \sqrt{x} \). This function is commonly used because it pairs each non-negative number with its principal square root.
One key aspect of \( \sqrt{x} \) is its domain. Because a square root of a negative number isn't a real number, \( \sqrt{x} \) only makes sense for \( x \geq 0 \) unless you're working within complex numbers.

• For example, in the given \(g(x) = \sqrt{1-x}\), the domain should satisfy the condition \(1-x \geq 0\) which simplifies to \(x \leq 1\).

Visually, the square root function starts at a point \( x = 0 \) or any valid lower boundary defined, like \( x = 1 \), and it increases gradually.
Rocketing straight from the origin on a graph, this smooth curve showcases that even small increases start to make the function grow faster.Understanding how the square root affects domains in composite functions is crucial, as seen when determining \(f(g(x))\) for the given exercise.

Quadratic Function

Quadratic functions are standard polynomial functions represented by the general formula \( f(x) = ax^2 + bx + c \). In the simplest form, \( f(x) = x^2 \), these functions create a parabolic graph opening upwards, or downwards in cases with negative coefficients.
Quadratics are known for their characteristic U-shape, with the vertex representing the function's maximum or minimum point.

• The domain of a simple quadratic function is all real numbers, \( x \in (-\infty, \infty) \), as there is no restriction on the inputs.
• However, when combined in function compositions like \(f(f(x))\), the form might become more complex such as \(x^4\), but the domain remains the same.

Analyzing how quadratics integrate within composite functions is vital. Observe how \(f(g(x))\) and \(g(f(x))\) incorporate quadratic aspects.
Within composite functions, the square root limits might influence or restrict the natural behavior of the quadratic components, resulting in a more limited domain.This exercise showcases how quadratic functions maintain their core domain yet must adapt to additional constraints in composite scenarios.