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Substitution is a fundamental technique used in mathematics to evaluate a function at a specific point. In the context of functions, substitution involves replacing the variable with a given number. For example, let's see how this works in our case with the function \(f(x) = 5x + 1\).To find \(f(-3)\), simply replace every occurrence of \(x\) in the function with \(-3\). This transforms the original function into a new expression: \(f(-3) = 5(-3) + 1\).

• The variable \(x\) is replaced by \(-3\).
• Every term in the function that includes \(x\) gets recalculated with this substitution.

This step sets the stage for further simplification using arithmetic operations. Understanding substitution helps decode functions and analyze expressions efficiently.

Once the substitution is complete, arithmetic operations are crucial for simplifying and solving the expression. Arithmetic operations include basic calculations like addition, subtraction, multiplication, and division. In evaluating functions, the order of operations matters greatly.In our example, after substituting \(x\) with \(-3\), we get the expression \(5(-3) + 1\). According to the order of operations (often remembered by the acronym PEMDAS: Parentheses, Exponents, Multiplication and Division, Addition and Subtraction), operations inside the parentheses are addressed first, but here we focus on multiplication.

• First, multiply: \(5 \times (-3) = -15\).
• Then add \(1\) to \(-15\) to address the addition operation: \(-15 + 1\).

These operations help in transitioning from a substituted expression to the final simplified result.

Simplification of expressions is the process of making them easier to understand or work with. It involves performing all possible arithmetic operations to reduce the expression to its simplest form.After performing substitution and the necessary arithmetic operations, you are left with an expression you've already partially simplified: \(-15 + 1\).

• Simply add these two numbers to further simplify: \(-15 + 1 = -14\).

It's essential to perform these operations carefully to ensure accuracy. Once the expression is fully simplified, you have effectively evaluated the function at that particular input value, arriving at \(f(-3) = -14\). This process not only helps solve the problem at hand but also enhances your overall problem-solving skills by providing clarity and simplicity.