Problem 63 - Algebra 2

Question

For each pair of functions, find \(f(g(x))\) and \(g(f(x))\) $$ f(x)=3 x, g(x)=x^{2} $$

Step-by-Step Solution

Answer

The result is \(f(g(x)) = 3x^2\) and \(g(f(x)) = 9x^2\).

Step 1: Calculate \(f(g(x))\)

First replace every place in the function \(f(x) = 3x\), where x occurs, with the function \(g(x) = x^2\). When \(x\) in \(f(x)\) is replaced with \(g(x)\), \(f(g(x)) = 3 * g(x) = 3x^2\). This means that the composite function \(f(g(x))\) is equal to \(3x^2\).

Step 2: Calculate \(g(f(x))\)

Now, replace the x in the function \(g(x) = x^2\) with \(f(x) = 3x\). Substituting \(f(x)\) into \(g(x)\) gives \(g(f(x)) = (f(x))^2 = (3x)^2 = 9x^2\). The composite function \(g(f(x))\) is thus equal to \(9x^2\).

Key Concepts

Function Composition

Function composition is like stitching two functions together. This means taking the output of one function and using it as the input for another. The notation used is typically either \(f(g(x))\) or \(g(f(x))\), which means that the result of function \(g(x)\) becomes the input for function \(f(x)\) and vice versa.

To understand better, think of it as a two-step process:

• Step 1: Calculate the inner function. Find \(g(x)\) or the first function outcome.
• Step 2: Input the result of the first function into the second function to get the final result.

In the original exercise, when calculating \(f(g(x))\), the function \(g(x)\) is first computed, being \(x^2\), and then its output is put into \(f(x) = 3x\), leading to \(3x^2\). For the reverse case, calculating \(g(f(x))\) involves squaring \(f(x) = 3x\) to get \(9x^2\). Composition is a powerful algebraic structure that allows blending multiple functional operations into one.

Algebra 2 Functions

Algebra 2 often explores functions in greater depth than basic algebra, focusing on different types of functions and transformations. It's not just about linear and quadratic functions but also encompasses composite functions and others like exponential and logarithmic functions.

Functions are essentially mathematical entities that produce output values for given input values. In this exercise, we have two functions, \(f(x) = 3x\) and \(g(x) = x^2\). At this level, understanding how to manipulate these expressions through techniques such as substitution is crucial. This involves replacing variables, simplifying expressions, and operating with different function forms.

• Linear Function: Represented by \(f(x) = 3x\), a simple multiplication of the input.
• Quadratic Function: Expressed by \(g(x) = x^2\), a square of the input.

Algebra 2 prepares students for more complex treatments involving matrices, polynomials, and complex numbers, building a strong foundation for higher mathematics study.

Squared Functions

Squared functions are functions where the variable \(x\) is raised to the power of two. Such functions create a parabolic shape graphically, which is symmetrical around the y-axis.

For example, the standard quadratic function \(y = x^2\) is a fundamental concept in mathematics, crucial in understanding the behavior of squared terms in functions.

• Basic Form: A simple \(x^2\) term results in a parabolic graph.
• Composite Squared Function: In function composition, a squared term can significantly transform another function, as seen when \(g(f(x)) = 9x^2\) in the original exercise.

Squaring a function often amplifies or alters its character significantly; thus, understanding its behavior is essential in tackling more complex algebraic expressions.

Through practice with squared functions, students gain insights into algebraic manipulations and functional transformations.