x g(x) yof g f(x) yof f input output input output Have to make sure that the output of g(x) = - 3. Domain: Find

Can you find another f and g. Hint: Which function is inside ( )’s? Remove the x3 + 5 and replace with x. g(x) Which expression is inside a grouping symbol? x3 + 5 Can you find another f and g. Hint: Which function is inside ( )’s? g(x) Which expression is inside a grouping symbol? x2 + 1

Not one-to-one. One-to-one function. Horizontal Line Test Both are functions. Vertical Line Test Vertical Line Test

f(x) = 2x + 3 g(x) = ½ ( x – 3 ) g( f(x) ) = ½ ( ( 2x + 3 ) – 3 ) The -1 is not an exponent. INVERSE = SWITCH ALL X & Y CONCEPTS! f(x) Inverse Identity x x y y f-1(x) Using the Inverse Identity, we need to show either f(g(x)) = x or g(f(x)) = x. f(x) = 2x + 3 g(x) = ½ ( x – 3 ) g( f(x) ) = ½ ( ( 2x + 3 ) – 3 ) f( g(x) ) = 2( ½ ( x – 3 ) ) + 3 g( f(x) ) = ½ ( 2x + 3 – 3 ) f( g(x) ) = ( x – 3 ) + 3 g( f(x) ) = ½ ( 2x ) f( g(x) ) = x Inverse Functions g( f(x) ) = x

Using the Inverse Identity, we need to show either f(g(x)) = x or g(f(x)) = x. Inverse Functions

Points: (-4, -3), (-2, 0), (4, 2) Points: (-3, -4), (0, -2), (2, 4) INVERSE = SWITCH ALL X & Y CONCEPTS! Switch the x and y coordinates. y = x 1. Replace f(x) with y. 2. Switch x and y. 3. Solve for y. Undo from the outside in. 4. Replace y with f -1(x). Points: (-4, -3), (-2, 0), (4, 2) Points: (-3, -4), (0, -2), (2, 4)

1. Replace f(x) with y. 2. Switch x and y. 3. Solve for y. Undo from the outside in. 4. Replace y with f -1(x). Domain Restriction y = x Vertex

1. Replace f(x) with y. 2. Switch x and y. 3. Solve for y. Undo from the outside in. 4. Replace y with f -1(x). Domain Restriction y = x Right side of parabola. Domain= Left side of parabola. Domain= Vertex

Domain Restriction. Tells us the we are looking at the right side of the parabola. This means a positive square root symbol. 1. Replace f(x) with y. 2. Switch x and y. 3. Solve for y. Undo from the outside in. 4. Replace y with f -1(x).

H.A. at y = 0 Common Point at ( 0, 1 ) b > 0 and b = 1 3-1= 1/3 H.A. at y = 0 *3 30= 1 Common Point at ( 0, 1 ) *3 31= 3 Domain: Range: *3 32= 9 b > 1: increase; 0 < b < 1: decrease r = 3 = b

I replaced BASE b with r to keep the letter b in our transformation rules. Negative, flip over the x-axis. 0 < |a| < 1, Vertical Shrink and |a| > 1, Vertical Stretch. Negative, flip over y-axis. 0 < |b| < 1, Horizontal Stretch and |b\ > 1, Horizontal Shrink. Solve for x. This is the Horizontal shift left or right. This is the Vertical shift up or down.

Translate Graph The negative will flip the graph over the x-axis. y = 0

Translate Graph The minus 2 is inside the function and solve for x. x = +2, shift to the right 2 units. y = 0

Translate Graph The minus 5 is outside the function and shift down 5 units. y = 0 y = -5

Translate Graph The negative on the x will flip the graph over the y-axis and solve 1 – x = 0 to determine how we shift horizontally. x = 1 y = 0

Translate Graph The plus 2 is inside the function and solve for x. x = -2, shift to the left 2 units. The minus 1 will shift down 1 unit. y = 0 y = -1

Translate Graph The negative on the x will flip over the y-axis. The negative in front of the 3 will flip the graph over the x-axis. y = 0

If bx = by, then x = y. Break down 8 One-to-one base 2 * 2 * 2 = 8 property. 3*3*3 3*3 Multiply powers Change to base 2 for all terms! 2*2*2*2*2 2*2 Multiply powers and use negative exponents to move 2 to the 5th up to the top. Mult. like bases, add exponents.

Manipulate to just 5x. Manipulate to just 3x. Substitute in 2 for 5x. Substitute in 4 for 3x.

Label the common point. Will be 1 unit away from the horizontal asymptote. Label the next point 1 unit along the x-axis that travels the y value away from the H.A. b = 4 } 1

Label the common point. Will be 1 unit away from the horizontal asymptote. Label the next point 1 unit along the x-axis that travels the y value away from the H.A. Up 3. Write as opposites! Down 3. Right 2. Flip the Common Point over the HA. Left 2. b = 4 Check the shifts to get the Common Point back to (0, 1)

Label the common point. Will be 1 unit away from the horizontal asymptote. Another approach. Label the next point 1 unit along the x-axis that travels the y value away from the H.A. Flip over x-axis. Left 2. Down 3. Start at (0,1) for the Common Point and flip over the x-axis. b = 4 Shift left 2 and down 3 to get to the common point in green.