Problem 29 - Algebra for College Students

Question

If \(f(x)=2 x-3\) and \(g(x)=x^{2}-3 x-4\), find \((f \circ g)(-2)\) and \((g \circ f)(1) . \quad 9 ; 0\)

Step-by-Step Solution

Answer

The compositions are 9 and 0, respectively.

Step 1: Compute (f ∘ g)(-2)

First find \(g(-2) = (-2)^2 - 3(-2) - 4 = 4 + 6 - 4 = 6\).
Then \(f(g(-2)) = f(6) = 2(6) - 3 = 9\).

Step 2: Compute (g ∘ f)(1)

First find \(f(1) = 2(1) - 3 = -1\).
Then \(g(f(1)) = g(-1) = (-1)^2 - 3(-1) - 4 = 1 + 3 - 4 = 0\).

Step 3: State the answers

\((f \circ g)(-2) = 9\) and \((g \circ f)(1) = 0\).

Key Concepts

Algebraic Functions

Algebraic functions are expressions made up of variables, constants, and arithmetic operations like addition, subtraction, multiplication, and division. These functions can also involve powers and roots. For example, given the functions \(f(x) = 2x - 3\) and \(g(x) = x^2 - 3x - 4\), both are algebraic functions because they combine basic algebraic operations.

• Linear Function: An example is \(f(x) = 2x - 3\). It represents a straight line and is described by a power of one.
• Quadratic Function: In the case of \(g(x) = x^2 - 3x - 4\), it's a quadratic function due to the square term \(x^2\), forming a parabola.

Understanding these functions helps in recognizing how different algebraic expressions behave when operations are applied. This foundational knowledge is crucial when performing operations on or compositions of these functions.

Function Operations

Function operations refer to different ways we can combine functions to form new ones. The main operations include addition, subtraction, multiplication, division, and composition. In this context, we focus primarily on function composition, which involves plugging one function into another.

• Addition/Subtraction: Given \( f(x) \) and \( g(x) \), you can form \( (f + g)(x) = f(x) + g(x) \).
• Multiplication/Division: Similarly, product \( (fg)(x) = f(x) \cdot g(x) \) or quotient \( (\frac{f}{g})(x) = \frac{f(x)}{g(x)} \), provided \( g(x) eq 0 \).
• Composition: Composition, indicated as \( (f \circ g)(x) \), requires evaluating \( g(x) \) first and then using that result in \( f(x) \).

In the provided exercise, the operation in focus is composition, exemplified by finding \( (f \circ g)(-2) \) and \( (g \circ f)(1) \). This requires us to intertwine two functions in a seamless manner.

Evaluating Functions

Evaluating functions involves substituting a given value into the function to find the result. It’s like solving a simple equation where you're given one variable to find the outcome. Here’s how you do it:

• Substitution: Plug the given value into the function wherever you see the variable \(x\).
• Solve: Carry out the arithmetic operations to get the answer.

For instance, with \(g(-2)\), you replace each \(x\) in \(g(x) = x^2 - 3x - 4\) with \(-2\), resulting in \((-2)^2 - 3(-2) - 4 = 4 + 6 - 4 = 6\). Similarly, evaluate \(f(1)\) in \(f(x) = 2x - 3\) by substituting \(1\) to get \(2(1) - 3 = 2 - 3 = -1\).
These evaluations help determine outputs for composed functions, leading to the correct answers for \( (f \circ g)(-2) \) and \( (g \circ f)(1) \) by moving step by step through the operations and calculations.