Welcome to an introduction to functions. A function is a mathematical relation that assigns each input exactly one output. We typically write functions using the notation f of x equals y. For example, the function f of x equals 2x plus 3 takes any input x, multiplies it by 2, and adds 3 to get the output. On the graph, we can see that when the input x is negative 1, the function gives us an output of 1. Every input value has exactly one corresponding output value, which is a defining characteristic of functions.
Let's explore different types of functions. Linear functions have the form f of x equals m x plus b, where m is the slope and b is the y-intercept. They create straight lines on a graph. Quadratic functions follow the pattern f of x equals a x squared plus b x plus c, forming parabolas. Exponential functions like f of x equals a to the power of x grow or decay at rates proportional to their current value. Logarithmic functions are the inverse of exponential functions. On our graph, we can see examples of linear, quadratic, and exponential functions. Notice how differently they behave: the linear function grows at a constant rate, the quadratic function accelerates as x increases, and the exponential function grows even more rapidly.
Understanding domain and range is crucial when working with functions. The domain is the set of all possible input values, or x-values, for which the function is defined. The range is the set of all possible output values, or y-values, that the function can produce. Let's look at the function f of x equals the square root of x as an example. Since we can't take the square root of a negative number in the real number system, the domain of this function is restricted to x greater than or equal to zero. Similarly, the square root of a positive number is always positive, so the range is also y greater than or equal to zero. On the graph, we can see that the function is only defined for x-values to the right of the y-axis, and all output values are positive.
Function transformations allow us to create new functions from existing ones. Let's explore some common transformations using the basic function f of x equals x squared. A vertical shift occurs when we add or subtract a constant. For example, f of x plus 2 shifts the parabola up by 2 units. A horizontal shift happens when we replace x with x minus h. The function f of x minus 1 shifts the parabola right by 1 unit. Vertical stretching or compression occurs when we multiply the function by a constant. And reflection across the x-axis happens when we negate the function, as in negative f of x. Notice how each transformation changes the graph in a specific way while maintaining its basic shape.
To summarize what we've learned about functions: A function is a mathematical relation that assigns each input exactly one output, typically written as f of x equals y. We explored various types of functions including linear, quadratic, exponential, and logarithmic functions, each with distinct behaviors and graphs. We discussed domain and range, which define the sets of valid inputs and possible outputs for a function. We also examined how functions can be transformed through vertical and horizontal shifts, stretches, compressions, and reflections. Functions are fundamental mathematical tools that help us model relationships between quantities in countless real-world applications, from physics and engineering to economics and data science.