Welcome to composite functions! A composite function is created when we combine two functions together. The key idea is that the output of one function becomes the input of another function. We write this as f of g of x, which means we first apply function g to x, then apply function f to that result.
Let's work through a concrete example. Given f of x equals x squared plus 1, and g of x equals 2x minus 3, we want to find f composed with g. First, we write f of g of x equals f of the quantity 2x minus 3. Then we substitute 2x minus 3 into the f function, giving us the quantity 2x minus 3 squared plus 1. Expanding this gives us 4x squared minus 12x plus 9 plus 1, which simplifies to 4x squared minus 12x plus 10.
It's important to understand that the order of composition matters. Function composition is not commutative. Using our previous example, we found that f composed with g equals 4x squared minus 12x plus 10. But if we reverse the order and find g composed with f, we get g of x squared plus 1, which equals 2 times the quantity x squared plus 1, minus 3. This simplifies to 2x squared minus 1. As you can see, f composed with g is not equal to g composed with f.
Let me summarize the key points about composite functions. First, they combine two functions where the output of one becomes the input of another. We write this as f circle g of x equals f of g of x. Remember to apply the inner function first, then the outer function. The order matters - f composed with g is generally not equal to g composed with f. Composite functions are widely used in calculus for the chain rule, in function transformations, real-world modeling, and computer programming.
Let's work through a concrete example. Given f of x equals x squared plus 1, and g of x equals 2x minus 3, we want to find f composed with g. First, we write f of g of x equals f of the quantity 2x minus 3. Then we substitute 2x minus 3 into the f function, giving us the quantity 2x minus 3 squared plus 1. Expanding this gives us 4x squared minus 12x plus 9 plus 1, which simplifies to 4x squared minus 12x plus 10.
It's important to understand that the order of composition matters. Function composition is not commutative. Using our previous example, we found that f composed with g equals 4x squared minus 12x plus 10. But if we reverse the order and find g composed with f, we get g of x squared plus 1, which equals 2 times the quantity x squared plus 1, minus 3. This simplifies to 2x squared minus 1. As you can see, f composed with g is not equal to g composed with f.
Let me summarize the key points about composite functions. First, they combine two functions where the output of one becomes the input of another. We write this as f circle g of x equals f of g of x. Remember to apply the inner function first, then the outer function. The order matters - f composed with g is generally not equal to g composed with f. Composite functions are widely used in calculus for the chain rule, in function transformations, real-world modeling, and computer programming.
Let's practice with a concrete problem. Given f of x equals 3x plus 2 and g of x equals x squared minus 1, find f composed with g evaluated at 2. First, we calculate g of 2, which equals 2 squared minus 1, giving us 3. Then we apply f to this result: f of 3 equals 3 times 3 plus 2, which equals 11. Therefore, f composed with g evaluated at 2 equals 11. Remember the key principle: always apply the inner function first, then the outer function.