Question

For the function defined as \(g(x)=\frac{x-2}{3}\), what is the value of \(g(-4)\) ? A) \(-2\) B) \(-\frac{2}{3}\) C) \(\frac{2}{3}\) D) 2

Short answer

A) \(-2\)

Step-by-step solution

Write down the function

Start with the given function: \[g(x) = \frac{x - 2}{3}\]

Substitute x with -4

Replace x with -4 in the function: \[g(-4) = \frac{-4 - 2}{3}\]

Simplify the expression

Solve the numerator (-4 - 2) and keep the denominator as is: \[g(-4) = \frac{-6}{3}\] Now divide -6 by 3: \[g(-4) = -2\] So, the value of \(g(-4)\) is -2. The correct answer is A) \(-2\).

Key concepts

Algebra

At its core, algebra is about finding the unknown or putting real life variables into equations and then solving them. In the context of our example, algebra is utilized to evaluate a function for a given value of its variable, which in this case is represented by the symbol 'x'.

When the function, such as our example's function defined as

\(g(x) = \frac{x - 2}{3}\),

needs to evaluate an unknown quantity, algebra allows for the manipulation of these symbols to discover the unknown value. A fundamental skill in algebra, therefore, is to be able to work with the symbolic representation of numbers and operations to manipulate and solve expressions and equations.

Substitution Method

The substitution method is a cornerstone of algebra, especially when it comes to functions. It is as simple as it sounds: you 'substitute' one value for the variable in the function.

For the function

\(g(x) = \frac{x - 2}{3}\),

the exercise requires finding the value of \(g(-4)\). This means that wherever there is an 'x' in the function, you replace it with '-4'. The substituted function will look like this:

\(g(-4) = \frac{-4 - 2}{3}\).

Applying the substitution correctly is crucial because it sets the stage for the simplifying and solving steps that follow.

Simplifying Expressions

Simplifying expressions is essentially the process of making them easier to understand and solve. In algebra, this often involves performing operations like addition, subtraction, multiplication, and division to reduce an expression to its simplest form.

In our example, once the substitution has been made, the next step is to simplify the expression

\(g(-4) = \frac{-4 - 2}{3}\)

to its simplest form. By adding the numbers in the numerator to get '-6' and keeping the denominator '3', we simplify the expression down to

\(g(-4) = -2\).

This final value is not just a solution; it's the outcome of carefully following algebraic procedures to reveal the specific value of a variable within a functional relationship.