Subset and functions

[StuckinMaths]

Let S be the subset of R² defined by
S={(x,y) ∈ R²: x ≤ 5, y ≥ 0, 2y ≤ x-1}
and let f and g be the functions defined by
f : R² → R²
(x,y) → (y,x)
and
g : R² → R²
(x,y) → (x-2, y+1)

a) On separate diagrams, sketch the set S and its image under each of the functions f and g.

b) Determine the composite function g ⁰ f, and sketch the image under g ⁰ f.

c) Show that g is one-one and onto, and determine its inverse function g⁻¹.

Please help! If you can upload pics or diagrams I would be most grateful.

 

[daon2]

Let S be the subset of R² defined by
S={(x,y) ∈ R²: x ≤ 5, y ≥ 0, 2y ≤ x-1}
and let f and g be the functions defined by
f : R² → R²
(x,y) → (y,x)
and
g : R² → R²
(x,y) → (x-2, y+1)

a) On separate diagrams, sketch the set S and its image under each of the functions f and g.

b) Determine the composite function g ⁰ f, and sketch the image under g ⁰ f.

c) Show that g is one-one and onto, and determine its inverse function g⁻¹.

Please help! If you can upload pics or diagrams I would be most grateful.

This is posted in advanced math, so let me ask you: Do you know how to graph lines in the xy plane? Graph inequalities? Can you explain in words to me what the functions f and g do geometrically?

 

[StuckinMaths]

The function f changes the gradient from m to 1/m.
g translates the graph 2 horizontally and 1 vertically.

I worked this out fine because I used y=2x to put the values of x in, to see how they change. But I am getting confused because I keep thinking I need an equation. With the function f I wasn't sure how to draw it, i understand how x and y change but just at a loss where to start with the drawing!

 

[daon2]

The function f changes the gradient from m to 1/m.
g translates the graph 2 horizontally and 1 vertically.

I worked this out fine because I used y=2x to put the values of x in, to see how they change. But I am getting confused because I keep thinking I need an equation. With the function f I wasn't sure how to draw it, i understand how x and y change but just at a loss where to start with the drawing!

You can't really 'draw' the graph of the function f, but you can draw its image. The function f is really a reflection along y=x, so yes, it will invert the slopes of lines. All vertical lines sent through f become horizontal ones. The region R = {(x,y) | x <= 5} becomes f(R) = {(x,y) | y<= 5}

\(\displaystyle g\circ f\) says "first do f, then do g." In words, it means reflect a point over the line y=x then do a translation. If it helps, these functions are also linear transformations which preserve your original "picture" S of your line y=1/2x-1/2. They just flip it over and move it a bit.

 

[mohammad]

subset

You can't really 'draw' the graph of the function f, but you can draw its image. The function f is really a reflection along y=x, so yes, it will invert the slopes of lines. All vertical lines sent through f become horizontal ones. The region R = {(x,y) | x <= 5} becomes f(R) = {(x,y) | y<= 5}

\(\displaystyle g\circ f\) says "first do f, then do g." In words, it means reflect a point over the line y=x then do a translation. If it helps, these functions are also linear transformations which preserve your original "picture" S of your line y=1/2x-1/2. They just flip it over and move it a bit.

could you help me understand this much clearly please

 

[mohammad]

cant open your message

[QUOTE
=mohammad;307696]could you help me understand this much clearly please[/QUOTE]
hi i cant open the message you sent to me could you please re send it back
thanks alot