Problem 23 - Algebra and Trigonometry with Analytic Geometry

Question

Exer. 21-34: Find (a) \((f \circ g)(x)\) and the domain of \(f \circ g\) and (b) \((g \circ f)(x)\) and the domain of \(g \circ f\). $$ f(x)=x^{2}-4, \quad g(x)=\sqrt{3 x} $$

Step-by-Step Solution

Answer

\((f \circ g)(x) = 3x - 4\), domain: \\(x \geq 0\\); \((g \circ f)(x) = \sqrt{3(x^2 - 4)}\), domain: \\(x \leq -2\\ or \\ x \geq 2\\).

Step 1: Understanding Function Composition

The task requires you to find two compositions: - \((f \circ g)(x)\) which means applying \(g\) first and then \(f\) to the result.- \((g \circ f)(x)\) which means applying \(f\) first and then \(g\) to the result.

Step 2: Find \\(f \\circ g\\)(x)\\

Start by substituting \(g(x)\) into \(f(x)\). Since \(f(x) = x^2 - 4\) and \(g(x) = \sqrt{3x}\), we have:\((f \circ g)(x) = f(g(x)) = f(\sqrt{3x}) = (\sqrt{3x})^2 - 4\)\This simplifies to:\(3x - 4\)

Step 3: Determine the Domain of \\(f \\circ g\\)(x)\\

The domain of \((f \circ g)(x)\) depends on the domain of \(g(x)\), which is the input for \(f(x)\). The function \(g(x) = \sqrt{3x}\) requires \(3x \geq 0\), thus \(x \geq 0\). The expression \(3x - 4\) has no further restrictions, so the domain of \((f \circ g)(x)\) is \([0, \infty)\).

Step 4: Find \\(g \\circ f\\)(x)\\

Substitute \(f(x)\) into \(g(x)\). For \(g \circ f\), we have:\(g(f(x)) = g(x^2 - 4) = \sqrt{3(x^2 - 4)}\)\This is the expression for \((g \circ f)(x)\).

Step 5: Determine the Domain of \\(g \\circ f\\)(x)\\

The expression under the square root \(3(x^2 - 4)\) must be non-negative:\[3(x^2 - 4) \geq 0\]Solving the inequality:\[x^2 - 4 \geq 0 \Rightarrow (x-2)(x+2) \geq 0\]The solution to this inequality is \(x \leq -2\) or \(x \geq 2\). Thus, the domain of \((g \circ f)(x)\) is \([-\infty, -2] \cup [2, \infty)\).

Key Concepts

Domain of a Function

Understanding the domain of a function is crucial for grasping how function composition works. The domain of a function refers to all the possible input values that the function can accept, ensuring the result is a real and valid number. In mathematical terms, the domain encompasses all real numbers for which the function is defined.

When we look at function compositions, such as \((f \circ g)(x)\) and \((g \circ f)(x)\), it's important to analyze them individually:

• For \((f \circ g)(x)\), the domain is dictated by the inner function \(g(x)\). Any restrictions on \(g\) directly influence the composite function because \(g\) provides inputs for \(f\).
• For \((g \circ f)(x)\), the situation is reversed. Here, \(f(x)\)'s outputs must be compatible with \(g(x)\)'s domain, further limiting the acceptable input values.

Each composite function has unique domain considerations based on these dependencies. Always consider each component within the composition chain to anticipate these domain restrictions.

Square Root Function

The square root function is a fundamental mathematical function written as \(\sqrt{x}\). This function outputs a number that, when multiplied by itself, returns the original input number. However, the square root function introduces limitations:

• The input of \(\sqrt{x}\) must be non-negative, as the square root of a negative number isn't a real number in standard arithmetic.
• This means, for any function containing a square root, the domain is restricted to where the radicand (the number or expression inside the square root) is zero or positive.

For practice with function compositions like \(g(x) = \sqrt{3x}\), investigate the condition \(3x \geq 0\). Here, \(x\) must satisfy this inequality, highlighting how the presence of a square root function restricts the domain to \([0, \infty)\).

These characteristics make square roots critical to consider when assessing and determining the domains of composite functions.

Quadratic Function

Quadratic functions are a staple in algebra, represented by the equation \(f(x) = ax^2 + bx + c\), where \(a\), \(b\), and \(c\) are constants. Their graph forms a parabola, either opening upwards or downwards.

• Quadratics have a domain of all real numbers, \(( -\infty, \infty)\), because squaring any real number results in a real number.
• Their outputs or range, however, depend on the parabola's direction and its vertex point, the highest or lowest point.

For instance, in solving \((f \circ g)(x)\) and \((g \circ f)(x)\), knowing \(f(x) = x^2 - 4\) informs us that \(f\) can take any real input and outputs values that depend on its quadratic nature. When considering \((g \circ f)(x)\), this quadratic's outputs become inputs to a square root, adding a layer of complexity to the domain. This highlights the significance of understanding how quadratics interact in function compositions to maintain valid inputs and ensure real outputs.