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

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

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

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

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

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

INVERSE = SWITCH ALL X & Y CONCEPTS! Switch the x and y coordinates. 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). 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). y = x Vertex Domain Restriction

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). y = x Vertex Domain Restriction Right side of parabola. Domain= Left side of parabola. Domain=

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. Tells us the we are looking at the right side of the parabola. This means a positive square root symbol.

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

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

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

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

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

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

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