3 feet in a yard
Let F(x,y) = y - x^3 Note that (-x)^3 = -(x^3) This suggests that F(-x,-y) = -F(x,y) (-x,-y) represents the point (x,y) reflected through the origin. You could say the function F has anti-point symmetry -- each point (x,y,F) is reflected through the origin at (-x, -y, -F).
22.5
f(x) defines a function of x. You can consider it to be y.
If ( f(x) = x - 3 ), then its inverse function can be found by swapping ( x ) and ( y ) and solving for ( y ). Setting ( y = x - 3 ) gives ( x = y - 3 ), or rearranging it, ( y = x + 3 ). Thus, the inverse function is ( f^{-1}(x) = x + 3 ). This means that applying ( f ) followed by ( f^{-1} ) (or vice versa) will return the original input.
It is 3.
Follow this example. f(x) = (x+3)/5 To find its inverse, write y=f(x) y= (x+3)/5 Interchange x and y x = (y+3)/5 solve for y in terms of x 5x=y+3 y=5x-3 The inverse of f(x) is f^-1(x) = 5x-3
Let F(x,y) = y - x^3 Note that (-x)^3 = -(x^3) This suggests that F(-x,-y) = -F(x,y) (-x,-y) represents the point (x,y) reflected through the origin. You could say the function F has anti-point symmetry -- each point (x,y,F) is reflected through the origin at (-x, -y, -F).
22.5
f(x) defines a function of x. You can consider it to be y.
If ( f(x) = x - 3 ), then its inverse function can be found by swapping ( x ) and ( y ) and solving for ( y ). Setting ( y = x - 3 ) gives ( x = y - 3 ), or rearranging it, ( y = x + 3 ). Thus, the inverse function is ( f^{-1}(x) = x + 3 ). This means that applying ( f ) followed by ( f^{-1} ) (or vice versa) will return the original input.
It is 3.
y=2x-3 y=-x-3
3 feet in a yard
That depends on the value of "y".
f(x) is the same thing as y= example: f(x)=2x+3 OR y=2x+3
1.5
Maybe; the range of the original function is given, correct? If so, then calculate the range of the inverse function by using the original functions range in the original function. Those calculated extreme values are the range of the inverse function. Suppose: f(x) = x^3, with range of -3 to +3. f(-3) = -27 f(3) = 27. Let the inverse function of f(x) = g(y); therefore g(y) = y^(1/3). The range of f(y) is -27 to 27. If true, then f(x) = f(g(y)) = f(y^(1/3)) = (y^(1/3))^3 = y g(y) = g(f(x)) = g(x^3) = (x^3)^3 = x Try by substituting the ranges into the equations, if the proofs hold, then the answer is true for the function and the range that you are testing. Sometimes, however, it can be false. Look at a transcendental function.