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Understanding and assessing linear functions is foundational in algebra. A linear function takes the form:

Understanding the Function

If given a function, such as \[ g(x) = 2x + 3 \], it simply means that for any input value of x, the output is calculated using this formula.

This skill is essential, as it helps in solving many real-world situations. Consider that the function describes a direct relationship between two quantities. Evaluating the given function for specific values involves replacing the variable with the given numbers and simplifying. Let's explore how to evaluate this function for different x-values.

Substitution is key to evaluating a function.

It involves replacing a variable (like x) with a specific number. For example, to find \( g(0) \), substitute 0 into the function wherever there's an x:
\[ g(0) = 2(0) + 3 \] Simplify to get: \[ g(0) = 3 \]

Similarly, to find \( g(1) \), replace x with 1: \[ g(1) = 2(1) + 3 = 2 + 3 = 5 \]

Also, to find \( g(-1) \), replace x with -1: \[ g(-1) = 2(-1) + 3 = -2 + 3 = 1 \]

Substitution allows you to evaluate functions at specific points by simply plugging in the values, making it easier to solve equations.

Algebraic operations play a crucial role in simplifying expressions after substitution. These operations include addition, subtraction, multiplication, and sometimes division. Let's look at how they apply in our function evaluation. When you substitute 0 into \( g(x) = 2x + 3 \), you perform a multiplication first: \[ g(0) = 2(0) \] Then, you add the constant term: \[ g(0) = 0 + 3 = 3 \] Simple yet fundamental operations help you arrive at the correct value.

For positive and negative substitutions, like 1 and -1, follow the same steps:

For \( g(1) \): \[ g(1) = 2(1) + 3 = 2 + 3 \] Which simplifies to: \[ g(1) = 5 \]

For \( g(-1) \): \[ g(-1) = 2(-1) + 3 \] Perform the multiplication first: \[ -2 + 3 \] Then the addition: \[ g(-1) = 1 \] Mastering these algebraic operations makes solving linear functions straightforward and ensures you always get the correct results.