Problem 60 - 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. $$(g \circ f)(0)$$

Step-by-Step Solution

Answer

The value of \((g \circ f)(0)\) is \(-21\).

Step 1: Identify the inner function

The input value is 0, which needs to be first plugged into the function \(f\). So evaluate \(f(0)\).

Step 2: Evaluate \(f(0)\)

Substitute the value \(0\) for \(x\) in the definition of \(f\), that is \(f(x)=2x-5\). Therefore, \(f(0)=2(0)-5=-5\).

Step 3: Identify the outer function

The result from the function \(f\) will be used as input for the function \(g\). So now evaluate \(g(f(0))\) which is same as \(g(-5)\).

Step 4: Evaluate \(g(-5)\)

Now, substitute the value \(-5\) for \(x\) in the definition of \(g\), that is \(g(x)=4x-1\). Therefore, \(g(-5)=4(-5)-1=-21\).

Key Concepts

Function Evaluation

Function evaluation is the process of determining the output of a function given a specific input. This involves substituting the input value into the function's equation and performing any necessary arithmetic operations. For example, if we have the function \( f(x) = 2x - 5 \), and we want to evaluate \( f(0) \), we substitute \( 0 \) for \( x \) and compute: \( f(0) = 2(0) - 5 = -5 \).

• Substitution: Replace the variable \( x \) with the given input value.
• Arithmetic operations: Perform the mathematical calculations based on the function's expression.

Evaluating a function is a fundamental step in understanding how inputs are transformed into outputs, paving the way for more complex operations like function composition.

Composite Functions

Composite functions combine two functions to form a new function. This involves applying one function to the result of another function.
Consider two functions, \( f(x) \) and \( g(x) \). The notation \( (g \circ f)(x) \) represents the composite function where \( f(x) \) is evaluated first, followed by \( g(x) \). Essentially, the output of \( f \) becomes the input to \( g \).

• Step 1: Compute \( f(x) \) using the given input value.
• Step 2: Use the output from \( f(x) \) as the input for \( g(x) \).

In our exercise, to find \( (g \circ f)(0) \), we first evaluate \( f(0) \) to get \(-5\), then evaluate \( g(-5) \) using \( g(x) = 4x - 1 \), leading to \(-21\). Composite functions are powerful tools for solving complex problems by breaking them down into simpler operations.

Algebraic Functions

Algebraic functions are functions that can be expressed using algebraic expressions. This includes operations like addition, subtraction, multiplication, division, and taking roots. These functions are ubiquitous in mathematics as they model relationships with polynomial and rational expressions.
For example, the function \( h(x) = x^2 + x + 2 \) is an algebraic function because it involves a polynomial expression. Algebraic functions like \( f(x) = 2x - 5 \) and \( g(x) = 4x - 1 \) are common in both pure and applied mathematics.

• Simple Polynomials: Algebraic functions can be simple, such as linear functions \( ax + b \).
• Complex Expressions: They can also include more complex polynomial expressions.

Understanding algebraic functions helps in evaluating and composing functions, which are essential skills in higher-level mathematics and various applications.