Problem 47 - Algebra 2

Question

If \(f(x)=2 x+4, g(x)=x-1,\) and \(h(x)=x^{2},\) find each value. $$ f[g(2)] $$

Step-by-Step Solution

Answer

f[g(2)] = 6

Step 1: Evaluate g(2)

Substitute 2 into the function \(g(x)\): \[g(2) = 2 - 1\] Calculate the expression: \[g(2) = 1\]

Step 2: Evaluate f[g(2)]

We need the value of \(f\) at \(g(2)\), which was found to be 1. Now substitute 1 into the function \(f(x)\):\[f(1) = 2 \times 1 + 4\] Calculate the expression: \[f(1) = 2 + 4 = 6\]

Key Concepts

Function Evaluation

Function evaluation is a fundamental part of mathematics, especially when working with algebraic functions. It involves finding the output value of a function for a given input. For example, in the exercise where the functions are given as \(f(x)=2x+4\), \(g(x)=x-1\), and \(h(x)=x^{2}\), we compute \(f[g(2)]\) by following these steps:

• First, evaluate \(g(2)\) by substituting 2 into \(g(x)\), which results in \(g(2) = 2 - 1 = 1\).
• Next, evaluate \(f(1)\) by substituting 1 into \(f(x)\), giving us \(f(1) = 2 \times 1 + 4 = 6\).

This method of function evaluation allows us to determine the specific value that a function will produce for any given input.

Algebraic Functions

Algebraic functions are expressions that involve algebraic operations such as addition, subtraction, multiplication, and division. These functions can include constants, variables, and exponents. In our example, we have three algebraic functions:

• \(f(x) = 2x + 4\), which is a linear function since it forms a straight line when graphed.
• \(g(x) = x - 1\), another linear function, simpler than \(f(x)\).
• \(h(x) = x^2\), a quadratic function that forms a parabola when graphed.

Each of these functions has its own unique structure and behavior based on these components. Understanding these properties helps to thoroughly analyze and calculate outputs for given inputs, enhancing comprehension.

Substitution Method

The substitution method is a powerful technique used in mathematics to simplify or solve equations by replacing one variable or function with another expression.

Using Substitution for Function Composition

In function composition, like \(f[g(x)]\), you perform substitution twice:

• First, substitute the input value into the inner function \(g(x)\) to get \(g(2) = 1\).
• Then, substitute this result into the outer function \(f(x)\), and evaluate it at \(f(1)\) to obtain the final result, \(f(1) = 6\).

This method is crucial in simplifying complex problems by breaking them down into manageable steps, making it easier to reach the solution without extensive rewriting.