Problem 21 - 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 \(g \circ f\). $$ f(x)=x^{2}-3 x, \quad g(x)=\sqrt{x+2} $$

Step-by-Step Solution

Answer

(a) \((f \circ g)(x) = x + 2 - 3\sqrt{x+2}\) with domain [-2, \infty)\; (b) \((g \circ f)(x) = \sqrt{x^2 - 3x + 2}\) with domain ((- \infty, 1] \cup [2, \infty)\).

Step 1: Understand the Composite Function

To find (a) \((f \circ g)(x)\), we first need to substitute the function \(g(x)\) into \(f(x)\). This means \((f \circ g)(x) = f(g(x))\). Similarly, to find (b) \((g \circ f)(x)\), we substitute \(f(x)\) into \(g(x)\), so \((g \circ f)(x) = g(f(x))\).

Step 2: Determine (f ∘ g)(x)

Let's compute \((f \circ g)(x)\):Substitute \(g(x) = \sqrt{x+2}\) into \(f(x)\):\[f(g(x)) = f(\sqrt{x+2}) = (\sqrt{x+2})^2 - 3(\sqrt{x+2})\]Simplify the equation:\[= x + 2 - 3\sqrt{x+2}\]

Step 3: Find the Domain of (f ∘ g)

To find the domain of \((f \circ g)(x)\), consider the domain of \(g(x)\) and ensure the substituted expression is in the domain of \(f(x)\).Since \(g(x) = \sqrt{x+2}\), \(x+2 \geq 0\).Thus, \(x \geq -2\) is the domain of \(g(x)\) and makes sure the expression inside the square root is non-negative.The domain of \(f(x)\) is \(\mathbb{R}\) (all real numbers), since there are no restrictions on the input values in \(f(x) = x^2 - 3x\).Hence, the domain of \((f \circ g)(x)\) is \([ -2, \infty)\).

Step 4: Determine (g ∘ f)(x)

Find \((g \circ f)(x)\):Substitute \(f(x) = x^2 - 3x\) into \(g(x)\):\[g(f(x)) = g(x^2 - 3x) = \sqrt{x^2 - 3x + 2}\].

Step 5: Find the Domain of (g ∘ f)

For the domain of \((g \circ f)(x)\), ensure the expression inside the square root is non-negative:Solve \(x^2 - 3x + 2 \geq 0\):Factor the quadratic:\[(x-1)(x-2) \geq 0\].The solutions are \(x = 1\) and \(x = 2\). Using a sign chart or test intervals, determine:\((-\infty, 1] \cup [2, \infty)\).Therefore, the domain of \((g \circ f)(x)\) is \((-\infty, 1] \cup [2, \infty)\).

Key Concepts

Domain of a Function

The domain of a function is the set of inputs that the function can accept. When dealing with real-valued functions, we're often interested in values of • that make sense in a given context, • don't result in undefined mathematical operations such as division by zero, or • don't result in imaginary numbers.

For example, in the square root function, like in the function \( g(x) = \sqrt{x+2} \), the expression inside the square root (often called the radicand) must be non-negative. Thus, for \( x+2 \geq 0 \) it implies that \( x \geq -2 \). Therefore, the domain of the square root function \( g(x) \) is all real numbers \( x \) such that \( x \geq -2 \). This ensures that all outputs of the square root are real numbers.

In function compositions, finding the domain becomes a bit more complex. For the composition \( (f \circ g)(x) \), the domain is restricted by \( g(x) \) and by the condition that \( g(x) \) must lie in the domain of \( f(x) \). This means considering the domains of both individual functions and how they interact.

Function Composition

Function composition involves creating a new function by applying one function to the results of another. Mathematically, function composition is denoted as \( (f \circ g)(x) = f(g(x)) \). It means you substitute the output of function \( g \) into the input of function \( f \).

U• nderstanding how to properly substitute one function into another is crucial in function composition. In our given exercise, to find \( (f \circ g)(x) \), we substituted \( g(x) = \sqrt{x+2} \) into \( f(x) = x^2 - 3x \). As a result, \( (f \circ g)(x) = (\sqrt{x+2})^2 - 3(\sqrt{x+2}) \). Simplifying gives \( x + 2 - 3\sqrt{x+2} \).• This substitution reveals that the composition can drastically change the form and behavior of the resulting function compared to the individual functions.

Another composition from the exercise is \( (g \circ f)(x) = g(f(x)) \), substituting \( f(x) = x^2 - 3x \) into \( g(x) \) to result in \( \sqrt{x^2 - 3x + 2} \). Each composition has a unique domain that reflects both its components.

Quadratic Function

A quadratic function is a type of polynomial function that can be written in the standard form \( f(x) = ax^2 + bx + c \), where \( a \), \( b \), and \( c \) are constants, and \( a \) cannot be zero. The graph of a quadratic function is a parabola, which can open upwards or downwards depending on the sign of the leading coefficient \( a \).

In our exercise, the function \( f(x) = x^2 - 3x \) is a quadratic function. Quadratic functions are defined for all real numbers, meaning their domain is \( \mathbb{R} \) (all real numbers).

These functions are important in finding compositions like \( (g \circ f)(x) \), where the input into \( g \) is this quadratic form. By solving equations like \( x^2 - 3x + 2 \geq 0 \), we can better understand where the composite function remains defined. This involves •finding roots and •using test intervals to find the domain, which we did using a sign chart or test values.

Square Root Function

The square root function is a mathematical function that involves taking the square root of its input. This function is generally written as \( g(x) = \sqrt{x} \) and has the constraint that \( x \geq 0 \), because there are no real square roots of negative numbers without introducing imaginary numbers.

In the function \( g(x) = \sqrt{x+2} \) from the exercise, it's critical for the radicand \( x+2 \) to be non-negative to ensure that the output of the square root is real. This gives us the domain condition \( x \geq -2 \).

When involved in function compositions, the constraints of square root functions influence the resulting domain significantly. For instance, employing \( g(x) \) in the composition \( (f \circ g)(x) = f(g(x)) \) means ensuring that not only is \( x+2 \geq 0 \), but the substituted expression \( f(g(x)) \) remains valid without introducing imaginary numbers or undefined operations.

Understanding how square root functions interact with other functions, especially in compositions, helps in accurately determining possible input values and highlights which intervals of \( x \) are permissible.