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Chapter 4
Section 2
One of the most important objects in mathematics is the function. A function is a special relationship between two sets. It assigns each element in the first set to a unique element in the second set.
Example 1. A seating assignment is a function. For each student, we assign them a specific seat. This seat is unique, since we cannot assign a student to sit in two different seats at the same time. Note: We could theoretically assign two different students to sit in the same seat at the same time. The seating assignment would still be a function, it just would not be practical.
Let’s look at the definition of a function.
Definition (Function). A function is a correspondence between two sets D and R such that for every x ∈ D it is assigned a unique y ∈ R.
Definition (Domain). The set D above is called the domain of the function. It is the set of values you can plug into the function.
Definition (Range). The set R above is called the range. It is the output values of the function.
Let see some examples and non-examples.
Example 2.
- y = x − 5 2. y = x^2 − 1 3. y =
x^2 − 5 x 4. y = ±
x
The function y = x^2 − 1 can be (and usually is) written using function notation:
f (x) = x^2 − 1.
So the name of this function is f.
Example 3.
- f (x) = x + 1 2. f (y) = y^2 − 11 3. f (x) = ln(x) − x
Sometimes we call the independent variable the argument of the function. Any letter can be used as the name of the function, except we cannot use the same letter as the variable.
Example 4.
- f (x) = x + 1 2. g(x) = x + 1 3. h(x) = x + 1 4. f 1 (x) = x + 1 5. φ(x) = x + 1
Suppose we have the function f (x) = x + 2. Then
f (1) = 1 + 2 = 3
f (−2) = −2 + 2 = 0
Example 5. Evaluate the function f (x) = 2x − 1
for
- x = − 2 2. x = 0 3. x = 10 4. x =
Example 6. Evaluate the function f (x) = x^2 + 2x − 1
for
- x = 0 2. x = − 2 3. x = a 4. x = a + h
Example 7. Let f (x) = x^2 − 2. Evaluate f (2) − 2 f (1) f (0) + 1
A very important expression is the difference quotient. We will use this when we study differential calculus.
Definition (Difference Quotient). The difference quotient is
f (a + h) − f (a) h
Example 8. Simplify the difference quotient if f (x) = x + 1.
Example 9. Simplify the difference quotient if f (x) = 2x − 3.
Example 10. Simplify the difference quotient if f (x) = x^2 + 1.
HW 4.2: 19-39 odd, EPS: difference quotient
Section ex^ and ln x
In addition to the algebraic functions we have looked at there are two more important functions we want to look at right now. Later we will look at the trig functions. They are the Exponential function, f (x) = ex, and its inverse function the natural log, f (x) = ln x. Recall that e is a number, e ≈ 2 .71828. Also the natural log, ln x, is the logarithm base e, ln x = loge x. Since the Exponential function and the natural log function are inverses of each other we can write the following:
ln ex^ = x , eln^ x^ = x
Your calculator has a button for both the Exponential function and the natural log function.
Example 14. Evaluate the following:
- e^0 =
- e^1 =
- e^2 =
- e^10 =
- e−^3 =
- e (^12) = 7. e.^3 = 8. ln 1 = 9. ln 2 =
- ln 3 =
- ln 15 =
- ln .2 =
- ln
- ln 0 =
HW EPS: ex,ln x
Section More Compositions, Decomposotions
Now more on compositions.
Example 15. Let f (x) = ex^ and g(x) = x^2 − 1. Find f (g(x)) and g(f (x)).
Example 16. Let f (x) = ln x and g(x) = x − 3. Find f (g(x)) and g(f (x)).
We can compose more than two functions at once.
Example 17. Let f (x) = ex, g(x) = x^2 − 1 and h(x) = ln x. Find f (g(h(x))) and h(g(f (x))).
When we start doing calculus it is important to be able to recognize compositions. So let us take a function and decompose it into its components. For example, suppose we have f (x) = e^2 x−^1. We would like to find functions g(x) and h(x) such that f (x) = g(h(x)). So for our example if we let g(x) = ex^ and h(x) = 2x − 1, then
f (x) = g(h(x)) = g(2x − 1) = e^2 x−^1.
There are certain functions that are typically used when we decompose a function. These are listed below:
- f (x) = anxn^ + an− 1 xn−^1 + · · · + a 2 x^2 + a 1 x + a 0
- f (x) =
x
- f (x) = n
x
- f (x) = ex
- f (x) = ln x
- f (x) = sin x
- f (x) = cos x
We will define the functions sin x and cos x a little later. I put them in the list for completeness.
Example 18. Decompose
- f (x) = e^2 x^ + 3ex^ − 1.
- f (x) = ln(x^2 − 5 x + 6).
Example 19. Decompose
- f (x) =
x^2 + 3
- f (x) = (x − 1)^2 + 11(x − 1) − 7.
Example 20. Decompose
- f (x) =
2 x − 4.
- f (x) =
x
Example 21. Decompose
- f (x) =
x + 1
- f (x) = e
√ (^3) x (^2) + .
HW EPS: more composition, decomposition
