Functions on the SAT: A Simple Guide

Welcome to our exploration of functions! Let's discover what makes a function special in mathematics.A function is a special mathematical relationship that follows one important rule.Think of a function like a machine. It takes an input, processes it according to specific rules, and produces exactly one output.When we put a number into our function machine, it follows the same process every time to give us our result.Let's look at a simple function that takes a number, multiplies it by two, and adds three.When we input two, the function multiplies it by two to get four, then adds three to get seven.With an input of three, we multiply by two to get six, add three, and get nine.And when we input four, we multiply by two to get eight, add three, and get eleven.Remember these important points about functions: Each input gives exactly one output. The same input will always give you the same output. And on the SAT, functions are typically written as f of x.Now that we understand what a function is, we're ready to learn more about how to work with them.On the SAT, functions are typically written using function notation.These different letters represent different functions, each taking an input and producing an output.Let's look at how to evaluate a function when given a specific input value.To find f of 3, we replace every x in the function with 3.First calculate three squared, which is nine.Then multiply two times three, which is six, and add one.Finally, add all terms to get sixteen.The SAT also tests function evaluation using variables. Let's see what happens when we substitute a plus h.Here's our function. We'll substitute a plus h for every x.When we make this substitution, we need to use the distributive property and FOIL method.After simplifying, we get this expanded form.Let's try a practice problem.Find g of negative two for this function.First, substitute negative two for x.Remember that negative two squared is positive four.Now multiply three times four, and don't forget that negative negative two is positive two.Finally, add all terms to get nineteen.Domain and range are fundamental concepts that define where a function can and cannot work.The domain represents all possible input values, or x-values, that a function can accept.The range consists of all possible output values, or y-values, that the function can produce.Let's look at our first example: f of x equals one over x. This function has an important domain restriction.The domain cannot include zero, because division by zero is undefined. This creates a vertical asymptote at x equals zero.Our second example is g of x equals the square root of x. Here, we can't input negative numbers.Our final example is the natural logarithm function. Its domain must be strictly positive numbers.On the SAT, you'll need to recognize these common domain restrictions and understand why they exist.On the SAT, you'll need to analyze different types of function graphs. Let's start with a linear function.For linear functions, identify the zero or x-intercept. Here, the zero occurs at x equals zero point five.Next, let's examine a quadratic function. Notice its parabolic shape and key features like the vertex and zeros.The absolute value function creates a V-shaped graph. Its vertex represents the minimum point.Functions can be transformed through shifts, stretches, and reflections. Here's a transformed quadratic function.Notice how the graph is shifted right by one unit and up by one unit, while also being stretched vertically by a factor of one half.When working with functions, we can combine them in several ways.The first way to combine functions is through basic arithmetic operations.Now, let's look at function composition, which is like nesting one function inside another.Let's work through a step-by-step example of function composition.Now, let's try a more challenging practice problem.First, we calculate h of three, which is nine minus one, giving us eight.Then, we find f of eight, which is eight plus one, giving us nine.