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Understanding how to evaluate functions is essential for students to navigate variables and equations efficiently. Function evaluation is finding the output of a function, given an input value. Consider a function as a machine that accepts an input, performs some operations on it, and gives an output. In our example, the function is defined as \(h(x) = 0.75x + 8\).

When we want to evaluate this function for different values of \(x\), we are simply feeding these values into our 'function machine' and observing what comes out. For instance, when \(x=2\), the output is \(h(2) = 9.5\). Similarly, we find that \(h(0) = 8\) and \(h(-3) = 5.75\). It's crucial for students to understand that the function remains the same, while the input values vary to produce different outputs.

The substitution method is the process we use to evaluate functions. It involves replacing the function's variable with a specific value and calculating the result. The main advantage of the substitution method is its simplicity and directness in assessing functions at various points.

In our exercise, to find \(h(2)\), we substitute \(x\) with 2 in the function \(h(x) = 0.75x + 8\), resulting in \(h(2) = 1.5 + 8\) after performing the arithmetic operations. The same steps apply for \(x=0\) and \(x=-3\), where we replace the variable \(x\) with 0 and -3, respectively, and compute the sums.

Points to Remember

• Always replace the variable with the provided value and enclose it in parentheses to avoid errors.
• Perform the arithmetic as it would naturally follow: multiplication before addition.
• Double-check your substitution to ensure accuracy before calculating.

Linear functions are the building blocks of algebra and form a straight line when graphed on the coordinate plane. They are characterized by the general form \(f(x) = mx + b\), where \(m\) is the slope and \(b\) is the y-intercept—the point where the line crosses the y-axis.

In our given function, \(h(x) = 0.75x + 8\), the slope is 0.75 signifying the steepness of the line, and 8 is the y-intercept, indicating that the line will cross the y-axis at (0,8). Linear functions show a constant rate of change between the variables, which explains why the output increases by 0.75 for every one unit increase in input.

Visual Interpretation

• Every linear function graph will be a straight line.
• The slope, positive or negative, dictates the direction in which the line slants.
• The y-intercept provides a starting point for the line on the graph.