Problem 10 - Algebra and Trigonometry with Analytic Geometry

Question

Exer. 9-10: Find (a) \((f \circ g)(x)\) (b) \((g \circ f)(x)\) (c) \((f \circ f)(x)\) (d) \((g \circ g)(x)\) $$ f(x)=3 x^{2}, \quad g(x)=x-1 $$

Step-by-Step Solution

Answer

(a) \(3x^2 - 6x + 3\), (b) \(3x^2 - 1\), (c) \(27x^4\), (d) \(x - 2\).

Step 1: Understanding Composition of Functions

The composition of functions is a process where you apply one function to the results of another function. In this exercise, you are calculating expressions like \((f \circ g)(x)\), which means you first apply \(g(x)\), then use the result as input for \(f(x)\). Similarly, \((g \circ f)(x)\) means apply \(f(x)\) first, then \(g\), and so on for other combinations.

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

For \((f \circ g)(x)\), we start by substituting \(g(x)\) into \(f(x)\). We know that \(g(x) = x - 1\) and \(f(x) = 3x^2\). Substitute \(x - 1\) into \(f\): \[ (f \circ g)(x) = f(g(x)) = f(x - 1) = 3(x - 1)^2 \]Now, expand the expression:\[ = 3(x^2 - 2x + 1) = 3x^2 - 6x + 3 \].

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

For \((g \circ f)(x)\), we apply \(f(x)\) first, then \(g\). So, substitute \(f(x) = 3x^2\) into \(g(x)\):\[ (g \circ f)(x) = g(f(x)) = g(3x^2) = 3x^2 - 1 \].

Step 4: Calculate \((f \circ f)(x)\)

This is the composition of \(f(x)\) with itself. Begin by substituting \(f(x) = 3x^2\) into itself:\[ (f \circ f)(x) = f(f(x)) = f(3x^2) = 3(3x^2)^2 = 3(9x^4) = 27x^4 \].

Step 5: Calculate \((g \circ g)(x)\)

For \((g \circ g)(x)\), use \(g(x)\) as the inner function and apply \(g\) to it again:\[ (g \circ g)(x) = g(g(x)) = g(x - 1) = (x - 1) - 1 = x - 2 \].

Key Concepts

Function Operations

Function operations involve performing mathematical processes that combine different functions together. They provide a way to manipulate functions to either add depth or create new mathematical expressions. There are several types of operations we can perform with functions:

• Addition and Subtraction: We can add or subtract functions to create a new function, represented as \( (f + g)(x) = f(x) + g(x) \) or \( (f - g)(x) = f(x) - g(x) \).
• Multiplication and Division: Functions can also be multiplied or divided, with division being defined wherever the divisor is not zero. This can become \( (f \times g)(x) = f(x) \times g(x) \) or \( \frac{f}{g}(x) = \frac{f(x)}{g(x)} \), respectively.
• Composition: One function applied to the result of another function, noted as \( (f \circ g)(x) = f(g(x)) \). This is a unique operation because the output of one function becomes the input of the next.

Understanding composition as a form of function operation is pivotal in calculus and algebra, as it is used to study more complex function behaviors and transformations.

Function Substitution

Function substitution is the key process involved in the composition of functions. It involves taking one function and replacing its variable with another function. This substitution allows us to compute the resulting transformation created by two or more functions working together.

When substituting, the inner function is calculated first. Consider the composition \( (f \circ g)(x) = f(g(x)) \). Here, the process is:

• First, calculate \( g(x) \), finding its result.
• Next, substitute that result into \( f(x) \), replacing every \( x \) in \( f \) with the value of \( g(x) \).

This step-by-step approach is crucial in ensuring each function component is correctly evaluated. By methodically substituting one function into another, we unveil how one function changes or transforms when driven through the modifications of another.

Polynomial Expressions

Polynomial expressions are mathematical phrases involving a sum of powers in one or more variables multiplied by coefficients. They appear frequently in function compositions. Understanding how to manipulate these expressions through operations like expansion and simplification is essential.

For instance, consider a quadratic polynomial \( ax^2 + bx + c \). When functions are composed, you may encounter expressions where one function is substituted into a polynomial. Take for example, in our exercise:

• When \( g(x) = x - 1 \) is substituted into \( f(x) = 3x^2 \), we get \( f(g(x)) = 3(x-1)^2 \).
• Expanding this, we get \( 3(x^2 - 2x + 1) = 3x^2 - 6x + 3 \).

The steps to expand these expressions include recognizing each term and applying algebraic rules. This manipulation gives a clearer picture of the function behavior and simplifies the complexity involved in calculations.

Function Transformation

Function transformations are techniques that modify the position, shape, or size of a graph of a function on the coordinate plane. Through transformations, we explore how function outputs change given certain inputs, which are common in compositions.

Transformations can include:

• Translations: Shifting the entire graph up, down, left, or right. For example, \( g(x) = x - 1 \) is a horizontal translation of \( f(x) = x \) by 1 unit to the right.
• Scaling: Stretching or compressing the graph vertically or horizontally.
• Reflection: Flipping the graph over a certain axis.

In compositions like \( (g \circ g)(x) = x - 2 \), you can see the effect of consecutive transformations. Each application of \( g \) shifts the result. Understanding these transformations aids in visualizing and predicting how composed functions behave, facilitating deeper comprehension of applied mathematics.