Math1000 section2.1
- 2. Math 1000 Stuart Jones What is a function? A function takes a domain D as input and outputs a unique element from R (the range) for each element of D.
- 3. Math 1000 Stuart Jones Evaluate f (3), f (−1), and f (0) if f (x) = −2x2 − 4x − 1
- 4. Math 1000 Stuart Jones Evaluate f (2), f (0) and f (a) if f (x) = √ 4x + 1
- 5. Math 1000 Stuart Jones The net change from a point x=a to a point x=b is the overall change in the y coordinate. It is represented as f (b) − f (a).
- 6. Math 1000 Stuart Jones Find the net change from x = 2 to x = 6 if f (x) = 2x2 −4x +2
- 7. Math 1000 Stuart Jones Find the net change from x = 0 to x = 10 if f (x) = 2−x x−11
- 8. Math 1000 Stuart Jones Piecewise functions are segments of different functions bound together into one function using a set of rules. For example: f (x) = x2 − 4 if x ≤ 0 2x + 1 if x > 0 This means that if x is 0 or less than 0, the top function will apply. However, if x is above 0, then the bottom function applies.
- 9. Math 1000 Stuart Jones Evaluate f (0), f (−1), and f (1) f (x) = x3 − 2x if x ≤ 0 3 √ x if x > 0
- 10. Math 1000 Stuart Jones Evaluate f (2), f (3), f (−1), and f (1) f (x) = 2 x if x ≤ −2 x + 4 if − 2 < x < 1 2−x x+4 if x ≥ 1
- 11. Math 1000 Stuart Jones Functions can also be evaluated at variables and other functions (you saw this earlier with f(a).) We will talk more about this in a future section, but if we have f (3x), all x terms get multiplied by 3, or if we have 3f (x), then the entire function gets multiplied by 3. Let’s look at a couple of examples.
- 12. Math 1000 Stuart Jones If f (x) = x−4 x+2, find f (−4x).
- 13. Math 1000 Stuart Jones If f (x) = √ x − 5, find 3f (x) − 2(f (x))2
- 14. Math 1000 Stuart Jones The last topic we will discuss in this section is one that will be important for anyone taking calculus later on. (And you still need to know it for this class anyway.) It is called the difference quotient. Difference Quotient For any function f (x), the difference quotient is defined as: f (a + h) − f (a) h NOTE: Problems involving the difference quotient will almost ALWAYS involve simplifying! If you still have an h in the denominator, you aren’t finished.
- 15. Math 1000 Stuart Jones Find the difference quotient of f (x) = 3x − 4
- 16. Math 1000 Stuart Jones Find the difference quotient of f (x) = 2x2 − 4x
- 17. Math 1000 Stuart Jones Find the difference quotient of f (x) = x x+2
- 18. Math 1000 Stuart Jones The Bottom Line Piecewise functions will return soon - make sure you understand them and how they work, or you’ll be very confused later when we have to graph them. Memorize the difference quotient formula. You’ll be using this later in this class and when you get to future classes.