Problem 59 Let $g(x)=3 x+2$ and \(f(x)=\f... [FREE SOLUTION]

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Chapter 7: Problem 59

Let $g(x)=3 x+2$ and $f(x)=\frac{x-2}{3} .$ Find each value. $$ g(f(-4)) $$

Short Answer

Expert verified

The value of the function composition $g(f(-4))$ is -4.

Step by step solution

Find the value of $f(-4)$

Substitute $x = -4$ into $f(x) = \frac{x - 2}{3}$.\nThus, $f(-4) = \frac{-4 - 2}{3} = \frac{-6}{3} = -2$.

Substitute $f(-4)$ into $g(x)$

Now, substitute $f(-4)$ which equals -2 into $g(x) = 3x + 2$.\nSubstituting, we get $g(f(-4)) = g(-2) = 3(-2) + 2 = -6 + 2 = -4$

Solution

The result of the function composition $g(f(-4))$ is -4.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Understanding Functions

In mathematics, a function is a fundamental concept that defines a specific relationship between two sets of elements, typically represented as inputs and outputs. A function assigns exactly one output to each input from its domain.
For example, if we consider the function $ g(x) = 3x + 2 $, $ x $ is the input variable, and $ g(x) $ is the output. Here, the expression $ 3x + 2 $ determines how $ g $ transforms the input $ x $ into an output.

Functions are often denoted by letters like $ f $, $ g $, and $ h $.
The input, typically $ x $, belongs to a set called the domain.
The output, which can be any form of expression, belongs to the codomain.

Functions can be linear, quadratic, or any type where a specific rule applies throughout its domain to produce a consistent result.

Introduction to Linear Functions

Linear functions are a specific type of function where the relationship between the input $ x $ and the output is a straight line when graphed. They have a general form $ f(x) = mx + b $. Here:

$ m $ is the slope of the line, which indicates the steepness and direction.
$ b $ is the y-intercept, where the line crosses the y-axis.

For example, in our original problem, the function $ g(x) = 3x + 2 $ is a linear function. The slope $ m $ is 3, signifying that for each increase by 1 unit in $ x $, the value of $ g(x) $ increases by 3. The y-intercept $ b $ is 2, meaning the line intersects the y-axis at the point (0,2).
Linear functions are easy to work with as they provide a constant rate of change second to none in simplicity compared to other functions like quadratic or exponential.

Exploring the Substitution Method

The substitution method is a vital technique in algebra used to solve equations or find the composition of functions. It involves replacing a variable with its known equivalent value, often derived from another equation or function.
In the context of our original exercise, we find $ g(f(-4)) $ by employing the substitution method. Here's how it works:

First, determine what $ f(-4) $ is by substituting $-4$ into $ f(x) = \frac{x - 2}{3} $. This gives $ \frac{-6}{3} = -2 $.
Next, use this result and substitute $-2$ into $ g(x) = 3x + 2 $, giving $ 3(-2) + 2 = -4 $.

Through substitution, we can find that $ g(f(-4)) = -4 $. The power of this method lies in simplifying many steps into manageable operations, making it easier to understand functions in terms of one another.

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91影视

Let \(g(x)=3 x+2\) and \(f(x)=\frac{x-2}{3} .\) Find each value. $$ g(f(-4)) $$

Short Answer

Step by step solution

Find the value of \(f(-4)\)

Substitute \(f(-4)\) into \(g(x)\)

Solution

Key Concepts

Understanding Functions

Introduction to Linear Functions

Exploring the Substitution Method

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