Problem 82 - Precalculus : Building Concepts and Connections

Question

In Exercises \(67-86,\) find expressions for \((f \circ g)(x)\) and \((g \circ f)(x)\) Give the domains of \(f \circ g\) and \(g \circ f\). $$f(x)=x-2 ; g(x)=2 x^{2}-x+3$$

Step-by-Step Solution

Answer

\(f \circ g(x) = 2x^2 - x + 1\) and \(g \circ f(x) = 2x^2 - 9x + 13\). The domains of both \(f \circ g\) and \(g \circ f\) are all real numbers.

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

The composite function \(f \circ g(x)\) is given by \(f(g(x))\). So substitute \(g(x)\) into \(f(x)\). Thus, you get \(f(g(x)) = g(x) - 2 = (2x^2 - x + 3) - 2 = 2x^2 - x + 1\). Therefore, \(f \circ g(x) = 2x^2 - x + 1\).

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

The domain of \(f \circ g\) is the set of all real numbers \(x\) such that \(g(x)\) is in the domain of \(f\). Since the function \(f\) is defined for all real numbers and \(g(x)\) = \(2x^2 - x + 3\) is a quadratic function (also defined for all real numbers), the domain of \(f \circ g\) is all real numbers.

Step 3: Find \(g \circ f(x)\)

The composite function \(g \circ f(x)\) is given by \(g(f(x))\). Substitute \(f(x)\) into \(g(x)\). Thus, \(g(f(x)) = g(x - 2) = 2(x-2)^2 - (x-2) + 3 = 2x^2 - 8x + 8 - x + 2 + 3 = 2x^2 - 9x + 13\). Therefore, \(g \circ f(x) = 2x^2 - 9x + 13\).

Step 4: Find the domain of \(g \circ f\)

The domain of \(g \circ f\) is the set of all real numbers \(x\) such that \(f(x)\) is in the domain of \(g\). Since both \(f\) and \(g\) are defined for all real numbers, the domain of \(g \circ f\) is all real numbers.

Key Concepts

Functions

A function is a mathematical relationship that uniquely maps each input to an output. Think of a function as a machine where you put an input into it, and it gives you an output based on some rule. For instance, the function \( f(x) = x - 2 \) subtracts 2 from any number \( x \) you put into it.
Functions are often denoted with letters like \( f \), \( g \), or \( h \), and they typically depend on one or more variables, usually \( x \) or \( y \).

• One important property of functions is that each input has exactly one output.
• Functions can be combined in various ways, such as with addition, multiplication, and composition.

Understanding the basic idea of functions is fundamental in all areas of mathematics.

Domain

The domain of a function is the set of all possible input values (usually values taken by \( x \)) for which the function is defined. In other words, it's the complete set of values that you can put into a function without causing problems like division by zero or taking the square root of a negative number.
In our exercise, we looked at composite functions, specifically \((f \circ g)(x)\) and \((g \circ f)(x)\). Composite functions involve replacing one function inside another.

• The domain of \( f \circ g \) is the set of all real number inputs that work for both \( g(x) \) and \( f(g(x)) \).
• Similarly, the domain of \( g \circ f \) consists of all real numbers that work for both \( f(x) \) and \( g(f(x)) \).

Determining the domain is crucial since it tells us what values our function can accept.

Quadratic Function

A quadratic function is a type of polynomial function where the highest power of the variable is 2. It takes the form \( ax^2 + bx + c \), where \( a \), \( b \), and \( c \) are constants and \( a eq 0 \).
Our example \( g(x) = 2x^2 - x + 3 \) is a quadratic function. The graph of a quadratic function is a parabola.

• If the coefficient \( a \) is positive, the parabola opens upwards, resembling a smile.
• If \( a \) is negative, it opens downwards, like a frown.

Quadratic functions are defined for all real numbers, which means their domain is all real numbers. Understanding this type of function is key in algebra and calculus.

Real Numbers

The real numbers include all the numbers you can think of, except complex numbers. This means it includes positive and negative numbers, whole numbers, fractions, and irrational numbers like \( \pi \) and \( \sqrt{2} \).
Working with real numbers is essential because they represent measurable quantities and can be found on a number line.

• Real numbers are utilized in many mathematical contexts, including functions, which often have domains restricted to all real numbers.
• In the exercise, both \( f(x) \) and \( g(x) \) are defined for all real numbers.

Understanding real numbers and their properties is fundamental to grasping more advanced mathematical concepts.