REVIEW: MORE FUNCTIONS AP CALCULUS :: MR. VELAZQUEZ

REVIEW: MORE FUNCTIONS AP CALCULUS :: MR. VELAZQUEZ

INVERSE FUNCTIONS Two functions are inverses if they undo each other. In other words, composing one function in the other will result in simply x (the independent variable) For example: f x = 5x 2 g x = x 5 + 2 5 (f g) x = 5 x 5 + 2 5 2 = x + 2 2 = x g f x = 5x 2 5 = 5x 5 2 5 + 2 5 = x + 2 5

INVERSE FUNCTIONS A function can only have an inverse when it is one-to-one, meaning that each value of x corresponds to one distinct value of y no repeated values. So officially, two one-to-one functions are inverses of each other if and only if: (f g) x = x AND g f x = x To find the inverse of a one-to-one function: 1. Replace f(x) with y. 2. Replace every x with y and every y with x. 3. Solve the equation for y. 4. Replace y with f 1 (x). 5. Verify by composing (f f 1 )(x) and/or (f 1 f)(x)

INVERSE FUNCTIONS Example: Find the inverse of each of the following functions. 1. f x = 2x 7 2. g x = 5x + 1 3. h x = x+4 2x 5

TRIG FUNCTIONS The trigonometric functions relate to the right triangles formed within a unit circle. cos θ, sin θ cos θ sec θ = 1 cos θ sin θ csc θ = 1 sin θ tan θ = sin θ cos θ cot θ = cos θ sin θ

TRIG FUNCTIONS We can also define the trig functions in terms of any right triangle by using SOHCAHTOA: sin θ = opposite hypotenuse cos θ = adjacent hypotenuse tan θ = opposite adjacent csc θ = 1 = hypotenuse sin θ opposite sec θ = 1 = hypotenuse cos θ adjacent cotθ = 1 tan θ = adjacent opposite

RADIAN ANGLE MEASURE In Calculus, angles are almost always indicated in radians Remember that a full circle is 2π radians and a semi-circle therefore is π radians Most angles will be sums or multiples of the angles in the table below

TRIG FUNCTIONS Examples: Evaluate each of the following. (a) cos 7π 6 (b) sin 13π 6 (c) sec 2π 3 (d) cot 5π 4 (e) csc 15π 2 (f) sec 22π 3

SOLVING TRIG EQUATIONS Often in Calculus, you will be asked to solve trigonometric equations, which may have a set of solutions rather than just one. Always note the interval over which you are asked to find solutions. More often than not, these will involve basic unit circle angles, so make sure you know those well! Examples: Find all solutions for the following trigonometric equations in the interval 0 t 2π (a) 2 sin t = 3 (b) 2 cos 3t = 2 (c) sin 2t = cos(2t)

INVERSE TRIG FUNCTIONS For certain situations, it may be necessary to find trig values for angles that aren t the standard unit circle angles we are used to. Solving these sort of trig equations requires the use of the inverse trig functions For these we can use a calculator, as long as we understand the domain and range of the inverse trig functions; if the desired angle is restricted to a quadrant outside the range, simply add or subtract π radians as needed.

INVERSE TRIG FUNCTIONS Example: Use a calculator to find the solutions for the following equations over their given intervals. (a) 4 sinθ = 3 π θ 2π (b) 12 cos t = 13 ( π t 0)

INVERSE TRIG FUNCTIONS Advanced Calculator Challenge: Use a calculator to find a set of solutions for each of the following trig equations. (You may use 3 decimal places) (a) 5 cos 2x 1 = 3 (b) 4 sin 2 t 3 3 sin t 3 = 1

EVEN AND ODD FUNCTIONS A function is even if f x = f(x). Even functions will always be symmetrical about the y-axis Examples: y = x 2, y = cos x A function is odd if f x = f(x). Odd functions will always be symmetrical about the origin. Examples: y = x 3, y = sinx EVEN ODD

EXPONENTIAL FUNCTIONS An exponential function is a function of the form f x = b x, where b > 0 and b 1 Some important properties of exponential functions: f 0 = 1. Any exponential function will take the value of 1 when x=0 f x f x 0. An exponential function will never be zero. > 0. An exponential function will always be positive. In other words, the range is f x 0, The domain is x, If 0 < b < 1, then f x 0 as x f x as x If b > 1 then f x as x f x 0 as x

THE NATURAL EXPONENTIAL The natural exponential function is f x = e x, where e = lim 1 + 1 n 2.718281828 n n This is an important function for calculus, so we will be spending a lot of time later dissecting this function. For now, simply know that e x and it s inverse ln x, are special functions in calculus.

CLASSWORK & HOMEWORK CLASSWORK 1. Write a short summary of what you learned today in your math journal 2. On a separate sheet of paper, for each type of function below, describe a realworld scenario that could be modeled by that sort of function, making sure to clearly define what the dependent and independent variables would be in each scenario 1. Linear Function 2. Quadratic Function 3. Exponential Function 4. Trig Function HOMEWORK Pg. 27, #1-30 Due 8/31