Problem 60 - College Algebra Essentials

Question

Find a. \((f \circ g)(x) \quad \) b. \((g \circ f)(x) \quad \) c. \((f \circ g)(2) \quad \) d. \((g \circ f)(2)\) $$f(x)=5 x-2, g(x)=-x^{2}+4 x-1$$

Step-by-Step Solution

Answer

a. \( (f \circ g)(x) = -5x^2 + 20x -7\), b. \( (g \circ f)(x) = -5x^2 + 20x -7\), c. \( (f \circ g)(2) = -4\), d. \( (g \circ f)(2) = -4\).

Step 1: Compose f(g(x))

Firstly, substitute g(x) into f(x) for \(f \circ g(x)\). Thus, it becomes \(f(g(x))= 5(-x^2 + 4x -1 ) -2\). This formula simplifies to \(-5x^2 + 20x -5 -2 = -5x^2 + 20x -7\).

Step 2: Compose g(f(x))

Next, substitute f(x) into g(x) for \(g \circ f(x)\). This results in \(g(f(x)) = - (5x-2)^2 + 4(5x-2) -1\). This simplifies to \( -5x^2 + 20x - 7\).

Step 3: Find (f ∘ g)(2)

Plug x = 2 into the function \(f \circ g(x)\) which we got in Step 1. It should be \(-5(2)^2 + 20*2 -7 = -4\).

Step 4: Find (g ∘ f)(2)

Finally, plug x = 2 into the function \(g \circ f(x)\) derived in Step 2. It should be \(-(10 -2)^2 + 4*(10 -2) -1 = -4\).

Key Concepts

Composite Functions

Understanding composite functions is a fundamental concept in algebra that deals with combining two different functions into one. When we compose two functions, say f(x) and g(x), we're essentially feeding the output of one function directly into the input of another. This is noted as (f \(\circ\) g) or f(g(x)), conveying that g(x) is applied first, and then f(x) is applied to the result.

The process can be imagined like a machine where for each input x, we first pass it through the machine g to obtain g(x), and then take that output and pass it through another machine, f, to obtain the final result. This cascading effect combines both machines' effects to yield a new, composite function.

Algebraic Functions

Algebraic functions are mathematical expressions constructed using algebraic operations, such as addition, subtraction, multiplication, division, and raising to a power, involving variables. In the example given from the exercise, f(x) = 5x - 2 and g(x) = -x^2 + 4x - 1 are both algebraic functions. They are defined in terms of 'x' and involve both polynomials and arithmetic operations.

An important property of algebraic functions is that they follow the same arithmetic rules we use for ordinary numbers, which allows us to manipulate and combine them through function composition. This results, in turn, in a new algebraic function that, while sometimes more complex, still abides by the same foundational algebraic principles.

Function Evaluation

Evaluating a function means finding the value of a function for a specific input. This concept is pivotal for understanding how functions behave and is often denoted by f(x), which signifies the function f's value at x. With composite functions, we must carefully evaluate the inner function before applying the outer function to its result.

For example, when given (f \(\circ\) g)(2), we evaluate g(2), then take that result and plug it into f(x). The process's accuracy is crucial since any mistakes made in the intermediate steps can lead to incorrect final outcomes. In our exercise, both instances of function composition at the given value resulted in the same output of -4, which reaffirms that when we're evaluating functions, performing each step methodically is vital, especially when dealing with composite functions.