Problem 59 - College Algebra Essentials

Question

Let $$\begin{aligned}&f(x)=2 x-5\\\&g(x)=4 x-1\\\&h(x)=x^{2}+x+2\end{aligned}$$ Evaluate the indicated function without finding an equation for the function. $$(f \circ g)(0)$$

Step-by-Step Solution

Answer

The result for \( (f \circ g)(0) \) is -7.

Step 1: Substitute x in function g

First, substitute \( x = 0 \) into the function \( g(x) = 4x - 1 \). This gives \( g(0) = 4(0) - 1 = -1 \).

Step 2: Substitute g(x) in function f

Then, substitute the result of \( g(0) \) into the function \( f(x) = 2x - 5 \). This gives \( f(g(0)) = f(-1) = 2(-1) - 5 = -7 \). This is the result of \( (f \circ g)(0) \).

Key Concepts

Composite Functions

Understanding composite functions is essential for diving deeper into the world of algebra. A composite function is the result of combining two functions in a specific order. It's like putting one function inside of another. The notation for a composite function is written as \( (f \circ g)(x) \), where two functions \( f(x) \) and \( g(x) \) are combined and \( f \) is applied to the result of \( g(x) \) when \( g(x) \) is evaluated first. The process can be visualized as a two-step journey. First, you input a value into the function \( g \) and get a result. That result then becomes the input for the function \( f \) to produce the final output.
Remember that the order in which you compose functions matters. The composite function \( (f \circ g)(x) \) will yield a different result from \( (g \circ f)(x) \) if \( f \) and \( g \) are different functions.

Evaluating Functions

Evaluating a function is simply about finding the output for a given input. When you're given a function, like \( f(x) = 2x - 5 \), the goal is to plug a specific value of \( x \) into the function and calculate the result. For instance, if you're asked to evaluate the function at \( x = 0 \), you'd substitute \( 0 \) for every instance of \( x \) in the function, leading to \( f(0) = 2(0) - 5 = -5 \).
When evaluating composite functions, you'll do this twice: once for each function involved. You first evaluate the inner function and then use its output as the input of the outer function. It's crucial to follow the steps in order and to remember that accuracy in each step ensures the correctness of the final result.

Algebraic Functions

An algebraic function is a type of function that can be expressed using algebraic operations, such as addition, subtraction, multiplication, division, and taking roots. In our exercise, the functions \( f(x) = 2x - 5 \), \( g(x) = 4x - 1 \), and \( h(x) = x^2 + x + 2 \) all represent algebraic functions. They are defined by polynomials, which are algebraic expressions consisting of variables and coefficients.
Understanding the behavior of these functions, such as the domain and range, how they increase or decrease, and their potential symmetries, is fundamental in algebra. Additionally, the operations used to combine these functions into new functions, as we do with composite functions, retain the nature of algebraic functions.

Function Operations

Just like you can add, subtract, multiply, and divide numbers, you can perform similar operations with functions. Function operations include addition, subtraction, multiplication, division, and composition of functions. When you add or subtract functions, you simply add or subtract their outputs for the same input value. Multiplication and division function operations combine outputs in a straightforward, number-like manner.
However, function composition, which is a keystone concept in algebra, is more complex. It combines the outputs of two different functions into a single new operation. Understanding function operations is crucial, as it allows you to build new functions from existing ones and explore their properties, which is particularly important in fields like engineering, computer science, and economics.