Problem 4 - Precalculus : Building Concepts and Connections

Question

Find \((f \circ g)(x)\) $$f(x)=\sqrt{x}, g(x)=(x+4)^{2}$$

Step-by-Step Solution

Answer

\(f(g(x)) =\sqrt{(x+4)^2}\)

Step 1: Identify the Functions

The two functions to consider are \(f(x) = \sqrt{x}\) and \(g(x) = (x+4)^2\)

Step 2: Define the Composition

The composition of the functions \(f\) and \(g\), denoted \(f \circ g\), means to apply the function \(g\) and then apply the function \(f\) to the result of that.

Step 3: Substitute \(g(x)\) into \(f(x)\)

Once \(g(x) = (x+4)^2\) is calculated, substitute this into \(f(x)\) to form \(f(g(x)) =\sqrt{(x+4)^2}\)

Key Concepts

Precalculus

Precalculus is the study of mathematical concepts that form the foundation for calculus, including functions, equations, and graphs. It often serves as a bridge between algebra and calculus, providing essential tools and knowledge for understanding rates of change, limits, and ultimately, derivatives and integrals, which are the core of calculus. In precalculus, students learn about different types of functions and their properties, and how to manipulate these functions algebraically and graphically. This is crucial, as it develops the analytical thinking required for more advanced studies in mathematics.

Composite Functions

Composite functions involve the combination of two or more functions such that the output of one function becomes the input of another. This is often noted as \((f \circ g)(x)\), which means you apply function \(g\), then take the result and apply function \(f\). Understanding composite functions is vital because it allows us to perform complex operations by breaking them down into simpler steps. The process of substituting one function into another to form a composite function is a fundamental concept in various mathematical fields, including calculus.

Square Root Function

The square root function, represented as \(f(x) = \sqrt{x}\), is crucial in mathematics as it answers the question: 'What number, when multiplied by itself, equals \(x\)?' It's characterized by its radical symbol and is the inverse operation of squaring a number. The square root function is part of a broader class of radical functions and it has a graph that forms part of a parabola in the first quadrant of a two-dimensional plane. Understanding the properties of square root functions is important for solving quadratic equations and analytically understanding distance and area in geometry.

Quadratic Function

A quadratic function is a second-degree polynomial, generally represented as \(g(x) = ax^2 + bx + c\), where \(a\), \(b\), and \(c\) are constants and \(a eq 0\). It forms a U-shaped curve known as a parabola when graphed. These functions are essential for modeling situations where there is acceleration, such as the path of a projectile in physics. Quadratic functions also appear in other disciplines, like economics and biology, where they model scenarios like revenue optimization and population growth. Learning to work with quadratic functions prepares students for a variety of problem-solving situations.