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Math 126 - Precalculus I Section 2.7 Combining Functions Algebra of Functions Let f and g be functions with domains A and B respectively. (f + g)(x) = f (x) + g(x)

Domain A \ B

(f  g)(x) = f (x)  g (x)

Domain A \ B

(f g)(x) = f (x)g (x) ✓ ◆ f f (x) (x) = g(x) g

Domain A \ B

Example 1. Let f =

p

x + 1 and g =

Domain {x 2 A \ B | g(x) 6= 0}

1 find following along with their domains. x4

(a) (f + g)(x) and (f + g)(3)

(c) (f  g)(x) and (f  g)(3)

(b) (f g)(x) and (f g)(3)

(d)

✓ ◆ ✓ ◆ f f (x) and (3) g g

Math 126 Lecture Notes

Section 2.7 Combining Functions

Composition of Functions Let f and g be functions, the composition of the functions is the following (f  g)(x) = f (g(x)) Example 2. Draw both the machine and arrow diagram representations of a function

Example 3. Let f =

p

x and g = x2 + 3 find following.

(a) (f  g)(x)

(c) (g  g)(x)

(b) (g  f )(x)

(d) (f  f )(x)

Instructor: Dr. Tim Meagher

Page 2 of 3

Math 126 Lecture Notes

Section 2.7 Combining Functions

1 Example 4. Let f = , g = x2  2 and h(x) = x + 3 find following. x (a) (f  g  h)(x)

(b) (h  g  f )(21)

Example 5. A ship is travel at 15 mi/h parallel to a straight shore line. The ship is 4 miles from the shore. It passes a light house at 2pm. Draw a picture that represents the distance to the lighthouse from the ship and the distance the ship traveled for any time passed 2pm. Express the distance from the light house a function distance travel down stream and the distance travel down stream as function of number of hours passed 2pm. Then compose the functions, what does this function represent?

Instructor: Dr. Tim Meagher

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Math 126 - Precalculus I Section 2.8 One-To-One Functions And Their Inverses Definition Of A One-To-One Function A function with domain A is called a one-to-one function if no two elements of A have the same image, that is, f (x1 ) 6= f (x2 ) whenever x1 6= x2

Example 1. Draw arrow diagram representations both a one-to-one function and not one-to-one function

Horizontal Line Test A function is one-to-one if and only if no horizontal line intersects its graph more than once. Example 2. Is the function one-to-one? (a) f (x) = x3

(c) f (x) = x2

(b) f (x) = x2 , wherex  0

(d) f (x) = 3x + 4

Math 126 Lecture Notes

Section 2.8 One-To-One Functions And Their Inverses

Definition Of The Inverse Of A Function Let f be a one-to-one function with domain A and range B. Then its inverse function f −1 has domain B and range A and is defined by f −1 (y) = x () f (x) = y for any y in B. Example 3. Draw arrow diagram representations a function and it inverse.

Inverse Function Property Let f be a one-to-one function with domain A and range B. The inverse function f −1 satisfies the following cancellation properties: f −1 (f (x)) = x 8x 2 A f (f −1 (x)) = x 8x 2 B

Example 4. Show that f (x) = x3 and g(x) = x1/3 are inverses of each other.

Instructor: Dr. Tim Meagher

Page 2 of 3

Math 126 Lecture Notes

Section 2.8 One-To-One Functions And Their Inverses

How To Find The Inverse Of A One-To-One Function • Write y = f (x) • Interchange x and y. • Solve this equation for y in terms of x (if possible). • Write y = f −1 (x) Example 5. Find the inverse of the following functions.

(a) f (x) = 4x  3

(b) f (x) =

2x+3 x−1

(c) f (x) =

x3 −2 6

(d) f (x) =

x+4 −3x−5

Graphing the Inverse of a Function The graph of f −1 is obtained by reflecting the graph of f in the line y = x. Example 6. Draw am example of this reflection

Instructor: Dr. Tim Meagher

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Math 126 - Precalculus I Section 3.1 Quadratic Functions and Models Quadratic Functions A quadratic function is a polynomial function of degree 2. f (x) = ax2 + bx + c,

a 6= 0

Standard Form of Quadratic Functions A quadratic function f (x) = ax2 + bx + c can be written in standard form. f (x) = a(x  h)2 + k This is done by completing the square. The vertex of the parabola (h,k) Example 1. Put the following in quadratic function into standard form by completing the square and then graph them. f (x) = 2x2  12x + 22

f (x) = 3x2  12x  10

Figure 1

Figure 2

Math 126 Lecture Notes

Section 3.1 Quadratic Functions and Models

Example 2. Put the quadratic function f (x) = ax2 + bx + c into standard form by completing the square and state the vertex.

Minimum or Maximum Value of Quadratic Functions The minimum or maximum value of quadratic function f (x) = ax2 + bx + c occurs at the vertex. b x=− 2a ✓ ◆ b If a > 0, then the minimum value is f − ✓ 2a ◆ b If a < 0, then the maximun value is f − 2a Note: You can use the vertex derived in Example 2 but it a lot to memorize so the above formulas work better in practice. Example 3. Find the minimum or maximum values for the following functions then graph them. f (x) = x2 + 4x

f (x) = −x2 + x + 2

Figure 3

Figure 4

Instructor: Dr. Tim Meagher

Page 2 of 3

Math 126 Lecture Notes

Section 3.1 Quadratic Functions and Models

Example 4. The gas mileage of car is based of the speed of the a given vehicle (assuming that many factors are constant). The following function is the miles per gallon (M) for a given speed (s) of a car. 1 2 s + 3s − 31 14 ≤ s ≤ 70 28 What is the best miles per gallon this car can achieve, and what speed does it occur? M(s) = −

Instructor: Dr. Tim Meagher

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Math 126 - Precalculus I Section 3.2 Polynomial Functions and Their Graphs Polynomial Functions A polynomial function of degree n is a function of the following form. an xn + an−1 xn−1 + ... + a1 x + a0 Where an , an−1 , ..., a 1 , a0 2 R and an 6= 0. Also n 2 N The numbers an , a n−1 , ..., a1 , a 0 are called the coefficients of the polynomial. The numbers a0 is the constant term. The number an is the leading coefficient and an xn is the leading term Example 1. Label all the above definitions in the following polynomial. 4x7 + 3x4 + 3x + 4

Example 2. Polynomial tend to be smooth and continuous, without cusp, corners breaks or wholes. Draw examples of functions that are and are not polynomials. Polynomial Functions

Not Polynomial Functions

Figure 1

Figure 2

Math 126 Lecture Notes

Section 3.2 Polynomial Functions and Their Graphs

End Behavior of Polynomials The end behavior of the polynomials P (x) = an xn + an−1 xn−1 + ... + a1 x + a0 is the by the degree n of the sign and the leading coefficient. Example 3. Draw examples of the end behavior polynomials Odd degree positive leading term

Odd degree negative leading term

Figure 3

Figure 4

Even degree positive leading term

Even degree negative leading term

Figure 5

Figure 6

Real Zeros of Polynomials If P is a polynomial and c is a real numbers, then following are equivalents • c is a zero of P • x = c is a solution of the equation P (x) = 0 • x  c is a factor of P (x) • c is an x-intercept of the graph P (x)

Instructor: Dr. Tim Meagher

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Math 126 Lecture Notes

Section 3.2 Polynomial Functions and Their Graphs

Intermedaite Value Theorem Polynomials If P is a polynomial functions and P (a) and P (b) have opposite signs then there most be a zero between a and b. Guildlines for Graphing Polynomials • Find the Zeros of the polynomial and marked these on the graph. • Test point between the zeros, and graph these • Determine the end behavior of the polynomial • Fill in the rest of the graph Example 4. Graph P (x) = (x  3)(x + 3)(x  1) by using the guidelines.

Table 1 Figure 7 x

P (x)

Instructor: Dr. Tim Meagher

Page 3 of 5

Math 126 Lecture Notes

Section 3.2 Polynomial Functions and Their Graphs

Example 5. Graph P (x) = x3  2x2  3x by using the guidelines.

Table 2 Figure 8 x

P (x)

Example 6. Graph P (x) = 2x4  x3 + 3x2 by using the guidelines.

Table 3 Figure 9 x

P (x)

Instructor: Dr. Tim Meagher

Page 4 of 5

Math 126 Lecture Notes

Section 3.2 Polynomial Functions and Their Graphs

The multiplicity of zeros and the graph The number of times the factor x  c is in a polynomial, is the multiplicity of that zero. If the multiplicity of zero is odd the graph will cross the x-axis and if it is even it will not. Example 7. Graph P (x) = x4 (x  2)3 (x + 1)2 by using the multiplicity of zeros and end behavior. Table 4 Figure 10 x

P (x)

The number of local extrema If a polynomial is of degree n, then graph of of the polynomial has at most n  1 local extrema. Example 8. For the given polynomial what is most n  1 local extrema can it have? (a) P (x) = x5 + 3x + 2

Instructor: Dr. Tim Meagher

(b) P (x) = 3x4  3x3 + 2

(c) P (x) = x2 + 3x + 2

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