Problem 15 - Algebra and Trigonometry with Analytic Geometry

Question

Exer. 11-20: Find (a) \((f \circ g)(x)\) (b) \((g \circ f)(x)\) (c) \(f(g(-2))\) (d) \(g(f(3))\) $$ f(x)=2 x^{2}+3 x-4, \quad g(x)=2 x-1 $$

Step-by-Step Solution

Answer

(a) \(8x^2 - 2x - 5\), (b) \(4x^2 + 6x - 9\), (c) \(31\), (d) \(45\).

Step 1: Understand the Composition of Functions

The composition of functions \(f \, \circ \, g\)(x) means you first apply function \(g\), and then apply function \(f\) to the result. Similarly, \(g \, \circ \, f\)(x) means applying function \(f\) first, then function \(g\).

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

Substitute \(g(x) = 2x-1\) into \(f(x)\). Therefore, \((f \circ g)(x) = f(g(x)) = f(2x-1)\).Replace \(x\) in \(f(x)=2x^2 + 3x - 4\) with \(2x-1\): \(= 2(2x-1)^2 + 3(2x-1) - 4\)Expand the function using the formula \(a^2 = (a+b)^2 = a^2 + 2ab + b^2\):\(= 2((4x^2 - 4x + 1) + 3(2x-1) - 4)\)\(= 8x^2 - 8x + 2 + 6x - 3 - 4 \= 8x^2 - 2x - 5\)Therefore, \(f(g(x)) = 8x^2 - 2x - 5\).

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

Substitute \(f(x) = 2x^2 + 3x - 4\) into \(g(x)\). Therefore, \((g \circ f)(x) = g(f(x)) = g(2x^2 + 3x - 4)\).Replace \(x\) in \(g(x) = 2x - 1\) with \(2x^2 + 3x - 4\):\(= 2(2x^2 + 3x - 4) - 1\)\(= 4x^2 + 6x - 8 - 1\)\(= 4x^2 + 6x - 9\)Thus, \(g(f(x)) = 4x^2 + 6x - 9\).

Step 4: Calculate \(f(g(-2))\)

First, evaluate \(g(-2)\):\(g(-2) = 2(-2) - 1 = -4 - 1 = -5\).Then, substitute \(-5\) into \(f(x)\):\(f(g(-2)) = f(-5) = 2(-5)^2 + 3(-5) - 4\)\(= 2(25) - 15 - 4\)\(= 50 - 15 - 4\)\(= 31\).Thus, \(f(g(-2)) = 31\).

Step 5: Calculate \(g(f(3))\)

First, evaluate \(f(3)\):\(f(3) = 2(3)^2 + 3(3) - 4\)\(= 2(9) + 9 - 4\)\(= 18 + 9 - 4\)\(= 23\).Then, substitute \(23\) into \(g(x)\):\(g(f(3)) = g(23) = 2(23) - 1\)\(= 46 - 1\)\(= 45\).Thus, \(g(f(3)) = 45\).

Key Concepts

Polynomial Functions

Polynomial functions are algebraic expressions made up of terms that include variables raised to whole-number exponents. The general form of a polynomial is given by\( a_nx^n + a_{n-1}x^{n-1} + ... + a_1x + a_0 \), where each \( a \) represents the coefficients. These coefficients can be any real number, and \( n \) can be any non-negative integer.

A polynomial function is characterized by its degree, which is determined by the highest power of the variable present in the expression. For example, the function \( f(x) = 2x^2 + 3x - 4 \) is a quadratic polynomial of degree 2 because the highest exponent of \( x \) is 2.

Polynomials are versatile tools in mathematics, and they can represent a wide variety of real-world phenomena, from the trajectories of objects to population growth models. Understanding polynomial functions is crucial for working with more advanced mathematical concepts.

Function Evaluation

Function evaluation involves determining the output of a function for a given input. This process is essential for understanding how functions work and for performing operations, like composing or combining functions.

To evaluate a function, substitute the input value into the expression and perform the necessary calculations. For instance, if you have a function \( f(x) = 2x^2 + 3x - 4 \) and you want to evaluate it at \( x = 3 \), you replace every \( x \) in the expression with 3. This results in the calculation \( f(3) = 2(3)^2 + 3(3) - 4 \), which simplifies to \( 23 \).

Accurate function evaluation helps verify the behavior of functions at specific values and provides insight into the function's graph and properties.

Algebraic Expressions

Algebraic expressions are combinations of numbers, variables, and operations (addition, subtraction, multiplication, division, and exponentiation) that represent mathematical relationships. They form the foundation of algebraic functions and mathematical modeling.

In the context of function composition, algebraic expressions play a fundamental role. For example, when composing functions \( f(x) \) and \( g(x) \), expressions from one function are manipulated using the operations of the other. This often involves substituting one entire algebraic expression into another, simplifying the result to form a new expression.

Understanding algebraic expressions is crucial for simplifying and solving mathematical problems, as they allow us to represent complex ideas in a manageable form.

Function Operations

Function operations are mathematical procedures that involve combining or manipulating functions. These operations include addition, subtraction, multiplication, division, and composition.

Function composition, in particular, involves applying one function to the results of another. Given two functions \( f(x) \) and \( g(x) \), the composition \( (f \circ g)(x) \) means applying \( g(x) \) first, then using \( f(x) \) on the result. This results in a new function formed by substituting the algebraic expression of \( g(x) \) into \( f(x) \).

The operations on functions allow us to build more complex functions from simpler ones, making them indispensable tools for mathematical analysis and problem-solving.