Chapter 5.1-5.2 Review (Operations on Functions and Inverses)

What is function notation?

Replacing the y with f(x) (ex. y=mx+b can be rewritten as f(x)=mx+b)

The sum of two functions, f and g

(f + g)(x) = f (x) +g(x)

The difference of two functions f and g

(f - g)(x) = f(x) - g(x)

The product of two functions f and g

(f*g)(x) = f(x)*g(x)

Let f(x)=x+2 and g(x)=5x Find (f*g)(x)

(f*g)(x)=5x^2 +10x

Let f(x)=x+3 and g(x)=x^2 +5x+6 Find (f+g)(x)

(f+g)(x)=x^2 +6x+9

The quotient of two functions f and g

(f/g )(x) = f(x)/g(x), If g(x) ≠ 0 (since you can never divide by 0)

Let f(x)=x+2 and g(x)=x+1 Find (f/g)(x)

(f/g)(x)= (x+2)/(x+1) but x+1 cannot = 0, therefore, x cannot =-1

The composition of two functions f and g

(fog)(x) = f(g(x)) or (gof)(x) = g(f(x))

f(x) = 3x + 4 and g(x) = 6x + 8, Find (f og)(x)

(fog)(x)= 3(6x+8) +4 = 18x+24+4 =18x+28

Steps for finding the inverse of a function are:

1) Set the function equal to y [ex. f(x) = y] 2) Swap the x and the y variables 3) Solve for y [Replace y with f^-1(x) once we know its an inverse function]

Find the inverse of f(x) = x + 7

1. y=x+7 2. x=y+7 3. x-7=y+7-7 x-7=f^-1(x)

Find the inverse of f(x) = (2/x) - 4

1.y=(2/x)-4 2.x=(2/y)-4 3.x+4=(2/y)-4+4 f^-1(x) = 2/(x +4)

Two functions are inverses iff (if and only if):

WAY 1= both their compositions are the identity function (aka (fog)(x) = x and (gof) = x ) WAY 2=if when you graph the functions and they are reflected over the line y=x (mirror image over y=x), their points from f(x)→g(x) go from (x,y)→(y,x)

Determine if the following two functions are inverses: f(x)=x+2 and g(x)=x-2

We can tell if they are inverse functions if (fog)(x) = x and (gof) = x (you can also graph them and see if they are reflected over the y=x): (fog)(x)= (x-2)+2 =x (Yes!) (gof)(x)= (x+2)-2 =x (Yes!) Since (fog)(x) = x and (gof) = x, f and g are inverses!

Determine if the following two functions are inverses: f(x)=4x+6 and g(x)=(x-6)/4

We can tell if they are inverse functions if (fog)(x) = x and (gof) = x (you can also graph them and see if they are reflected over the y=x): (fog)(x)= 4((x-6))/4+6 = x-6+6 =x (Yes!) (gof)(x)= (4x+6-6)/4 = (4x)/4 =x (Yes!) Since (fog)(x) = x and (gof) = x, f and g are inverses!

What is an inverse function? (f^-1)(x)

f f(g(x)) = x and g(f(x)) = x, then f and g are Inverses and (f^-1)(x) is said "f is an inverse function of x)