Problem 38 - Algebra 2

Question

If \(f(x)=4 x, g(x)=2 x-1,\) and \(h(x)=x^{2}+1,\) find each value. $$ g[g(7)] $$

Step-by-Step Solution

Answer

The value of \(g[g(7)]\) is 25.

Step 1: Evaluate Inner Function

First, evaluate the inner function, which is \(g(7)\). Substitute 7 into the function definition \(g(x) = 2x - 1\). Calculate as follows: \[ g(7) = 2(7) - 1 = 14 - 1 = 13. \] So, \(g(7) = 13\).

Step 2: Evaluate Outer Function

Next, use the result from Step 1 to evaluate \(g(13)\). Substitute 13 into the function \(g(x) = 2x - 1\). Calculate as follows: \[ g(13) = 2(13) - 1 = 26 - 1 = 25. \] So, \(g(13) = 25\).

Key Concepts

Evaluation of Functions

Evaluating a function involves plugging a specific number into the function's formula where the variable is located. Let's illustrate this with the function \( g(x) = 2x - 1 \), which asks us to evaluate it at a particular value of \( x \). To do this, replace \( x \) with the given number and perform the arithmetic operations indicated by the function.

For example, to evaluate \( g(7) \), substitute 7 into the function:

• First, multiply 7 by 2, which gives 14.
• Then subtract 1 from 14, resulting in 13.

So, \( g(7) = 13 \).
This process provides the output of the function for the input of 7, proving essential for understanding how the function transforms values.

Nested Functions

Understanding nested functions is about evaluating functions where one function is located inside another. This process occurs step-by-step and requires us to find the result of an inner function before using its output as the input for the outer function.

In the case of evaluating \( g[g(7)] \), we perform the operation in stages:

• First, find the result of the inner function \( g(7) \), which yields 13.
• Next, use the 13 obtained from \( g(7) \) as the input for the outer function \( g(x) \).
• Finally, evaluate \( g(13) \) to get the ultimate result of 25.

This process, akin to peeling an onion layer by layer, is fundamental in dealing with more complex algebraic problems that involve multiple levels of functions.

Basic Algebra Calculations

Algebraic calculations often form the backbone of function evaluations, comprising straightforward arithmetic operations. Let's break down the necessary operations for evaluating \( g(7) \) and \( g(13) \) using simple steps.

For both evaluations involving the function \( g(x) = 2x - 1 \):

• Multiplication: Multiply the given number by 2.
• Subtraction: Subtract 1 from the result obtained after multiplication.

Specifically,

• For \( g(7) \), it was calculated as \( 2 \times 7 - 1 = 13 \).
• Following the same process, \( g(13) \) became \( 2 \times 13 - 1 = 25 \).

Basic operations like these are foundational to understanding algebra, as they repeat in nearly every problem involving calculations.