5.1 Polynomial Functions Interval Notation: Sets of real numbers that make up portions of the number line can be represented by intervals. A closed interval, [a; b] = fx j a ≤ x ≤ bg; the set of all number between a and b, including the endpoints. An open interval, (a; b) = fx j a < x < bg; the set of all numbers between a and b, NOT including the endpoints. An interval can also be half-open, half-closed: such as (a; b] or [a; b). Parentheses indicate that the endpoint IS NOT included. Square brackets indicate that the endpoint IS included. The union of two intervals A [ B is the set of all numbers in either interval. Examples: Represent the following using interval notation. All real numbers −3 < x ≤ 7 x > −3 and x ≤ 7 All x where x 6= 7 x ≥ 3 and x 6= 8 x < 2 and x 6= −5; −2 x ≤ −3 or x > 5 1 What is a function? A function is a rule that assigns to each input value exactly one output value. We write y = f(x) and say \y is a function of x" to indicate that f takes values of x and assigns them values of y. The \input" variable is called the independent variable and the \output" variable is called the dependent variable. Example: For the function f(x) = 3x2 + 2x, find: f(1) f(−3) f(a) f(a + h) f(a + h) − f(a) The values you get when you \plug in" numbers into the function are the y-values in the graph. Vertical Line Test: A graph in the coordinate plane is the graph of a function if and only if there is no vertical line that intersects the curve more than once. In other words, if any vertical line crosses a graph more than once, it is not a function. Are the following graphs of functions? 2 The domain of a function is the set of all x-values for which the function is defined. The range of a function is the set of all y-values that the function attains or takes on. For the graphs of the following functions, find f(−2); f(0); f(1) and then find the domain and range of each function. 3 A polynomial function is a function of the form n n−1 f(x) = anx + an−1x + ··· + a1x + a0 3 2 3 Examples of polynomials: f(x) = x , f(x) = 2x − 6x + 7, f(x) = 2 x − 4 p 1 1=2 3 NOT polynomials: f(x) = x , f(x) = x , f(x) = x The degree of the polynomial is the highest power of x that appears. The numbers a0; a1; : : : ; an are called the coefficients of the polynomial. The number an, the coefficient of the highest power of x, is called the leading coefficient and the term n anx is called the leading term. The number a0 is called the constant coefficient or constant term. Examples: For the following polynomial functions, state the degree of the polynomial, the leading term, the leading coefficient, and the constant term. f(x) = −3x4 − 2x + 7 g(x) = −3x2 + 5x3 h(x) = 6x − 9 Polynomials of degree 2 are called quadratic functions. Polynomials of degree 3 are called cubic func- tions. You should be comfortable with recognizing the graphs of the parent functions f(x) = x2 and f(x) = x3 shown below. Graphs of other quadratic and cubic polynomial functions are just shifts and stretches and reflections of these two parent functions. The domain of all polynomial functions is all real numbers: (−∞; 1). The range depends on the polynomial. 4 Definition: The zeros of a function f(x) are the values of x such that f(x) = 0. In other words, they are the x-intercepts of the function. What are the zeros of the function graphed below? Finding zeros usually requires factoring!! If you don't remember how to factor, though, you can ALWAYS use the quadratic formula. The solution(s) to the equation ax2 + bx + c = 0 are given by p −b ± b2 − 4ac x = provided b2 − 4ac ≥ 0 2a Find the zeros of the following functions. f(x) = 4x2 − 36 f(x) = x2 + 4x − 21 f(x) = −6x2 + 14x − 4 f(x) = 2x3 − 5x2 − 12x 5 f(x) = x2 − 3x − 3 The graph of EVERY quadratic function is a parabola. Whether the parabola opens upward or downward depends on the sign of a. The vertex of a parabola is the point on the graph that has either the lowest y-value (when opening up) or the highest y-value (when opening down). b If a quadratic function is written in standard form, the x-coordinate of the vertex is x = − . 2a b Then, the y-coordinate of the vertex, is just f − 2a Example: Consider the quadratic function f(x) = 3x2 − 12x + 15. • Find the vertex and determine if the parabola opens upward or downward. • What is the minimum value? maximum value? • What are the zeros? 6 Example: Consider the quadratic function f(x) = −4x2 − 12x − 8. • Find the vertex and determine if the parabola opens upward or downward. • What is the minimum value? maximum value? • What are the zeros? Applications of Quadratics In Chapter 1, we saw that when the selling price s of an item is fixed, then the revenue function is R(x) = sx. We also learned, however, that sometimes the price p of an object depends on the quantity demanded by consumers x and is given by the demand equation. So, in general, if we are given a demand equation p = ax + b, then the revenue function will now be R(x) = px = (ax + b)x = ax2 + bx, which is a quadratic function. Example: Suppose the price-demand equation for a product is given by p = −7x + 84. • What is the revenue function? • How many items should be sold to maximize revenue? What is the maximum revenue? • What is the price per unit when the revenue is maximized? • Now suppose the cost function for this company is C(x) = 35x + 70. How many items does the company need to sell to break-even? 7 Example: A company sells gadgets. They can sell 10 gadgets when the price is $170 and they can sell 20 gadgets when the price is $120. The company incurs production costs of $70 per gadget and the company has fixed costs of $805. • Find the price-demand equation, assuming it is linear. • How many items should be sold to maximize revenue? What is the maximum revenue? • At what price are the gadgets sold when revenue is maximized? • What is the company's profit function? • How many items should be sold to maximize profit? What is the maximum profit? • Determine the price of a gadget when the profit is maximized. • How many gadgets does the company need to sell to break even? 8 5.2 Rational, Power, and Piecewise Functions Rational Functions p(x) A rational functions is a function of the form f(x) = q(x) where p(x) and q(x) are polynomials. The only place a rational function is NOT defined is where the denominator is zero!! So, the domain of a rational function is all real numbers EXCEPT the values of x for which the denominator is 0. Find the domains of the following rational functions. x + 3 • f(x) = x2 + 7x − 8 x(x + 4) • f(x) = x3 − 16x To add/subtract rational expressions or functions, remember you must get a common denominator. x − 3 x + 5 Add and simplify + x + 2 x − 1 9 f(a + h) − f(a) The difference quotient is the quantity This quantity is very important for calculus. h Examples: Find and simplify the difference quotient for the following functions. • f(x) = −x2 − 3x + 1 2 • f(x) = x − 1 10 Power Functions and Radical Functions A power function is a function of the form f(x) = axr where r is a real nonzero number. Functions such as f(x) = x2 and f(x) = x3 are power functions as well as polynomial functions. But since r does not have to be an integer in a power function, we now have new functions like f(x) = x1=2 and f(x) = x1=3 which are power functions but not polynomial functions. If a power function has a fractional exponent, then it can also be referred to as a radical function. The reason why is the following imporant relationship: p p x1=n = n x xm=n = n xm So fractional exponents are radicals. x1=2 = x1=3 = x3=4 = p p The graph of f(x) = x1=2 = x and f(x) = x1=3 = 3 x are shown below. These are two more parent graphs you should be familiar with. For a function that involves radicals, the key to finding the domain is realizing that you can not take an even root of a negative number!! Whatever you are taking an even root of must be ≥ 0. If the even root is in the denominator, then it must be > 0. For odd roots, since you CAN take the odd root of a negative number, there are no domain issues with odd roots, unless it is in the denominator, in which case it cannot be 0. 11 Find the domains of the following functions. p • f(x) = 3x − 8 p • f(x) = 3 5x + 9 x • h(x) = p 4 −6x − 5 p x + 1 • g(x) = (x − 4)(4x + 3) p 4 3x − 1 • f(x) = p 3 3x − 4 12 To rationalize the numerator or denominator of a fraction that involves radicals means to get rid of any radicals in that part of the fraction.