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A linear function is one of the simplest types of functions you'll encounter in algebra. It is defined by a formula of the form \( h(x) = ax + b \), where \( a \) and \( b \) are constants, and \( x \) represents the input variable. This kind of function graphs as a straight line and is characterized by its constant rate of change or slope, indicated by the coefficient \( a \). The constant \( b \) is known as the y-intercept, which is the point where the line crosses the y-axis.

In our exercise, the given function \( h(x) = 3x + 6 \) is a linear function with a slope \( a = 3 \) and y-intercept \( b = 6 \). This means for every unit increase in \( x \), \( h(x) \) increases by 3 units. Understanding the structure of a linear function is crucial because it allows us to predict how the function behaves over different input values.

The substitution method is a straightforward and effective way to evaluate functions. When a problem asks you to "evaluate a function," it means to determine the function's output for specific input values. You do this by substituting the given value of \( x \) into the function, then simplifying to find \( h(x) \).

Here's how you go about it:

• Identify the value of \( x \) you need to substitute.
• Replace every instance of \( x \) in the function with this value.
• Perform the arithmetic operations to simplify the expression and find the result.

For example, to evaluate \( h(x) = 3x + 6 \) at \( x = 2 \), substitute \( 2 \) for \( x \), yielding \( h(2) = 3(2) + 6 = 12 \). Repeat the same steps for other values of \( x \), as shown in the exercise's solutions.

An algebraic expression is a combination of numbers, variables (like \( x \)), and arithmetic operations (such as addition or multiplication). They can be simple, like \( 3x \), or more complex, like \( 3x + 6 \). Understanding how to work with algebraic expressions is key to solving algebra problems.

In the expression \( 3x + 6 \), \( 3x \) is known as a term, with \( 3 \) being the coefficient of \( x \). The expression represents the output of the function for any given \( x \). When you substitute a value for \( x \), you convert the expression into a single numerical value, utilizing basic arithmetic operations.
Mastering skills with algebraic expressions involves learning to combine like terms, apply the distributive principle, and perform substitutions correctly, all of which are vital for evaluating functions or solving equations.