help with precalculus

Question

1/ Determine if the function is one-to-one.

 

 

i think yes

2/ Confirm that f and g are inverses by showing that f(g(x)) = x and g(f(x)) = x.



f(x)=4/x and g(x)=4/2

Answer 1

1 fails the horizontal line test. It's not one to one. You can draw a horizontal line and cut the graph in two places. To be one to one, it has to impossible to cut the graph horizontally or vertically in more than 1 point.

2 f(x) = 4/x, g(x) = 4/2 = 2

f(g(x)) = f(2)= 4/2 = 2

g(f(x)) = g(2) = 2 ( g(anything) = 2)

they're inverses. you start with 2, and end with 2 and vice versa. For inverses, if you start with a number, the inverse brings you back to the same number.

inverse of f(x) is found by switching x and y

y=4/x switch them

x=4/y solve for y

y=4/x. f(x) = 4/x = f^-1(x) f(x) is it's own inverse

g(x) = 2 is y=2. draw a 45 degree line through the origin. the inverse is the mirror image on the other side of that 45 degree diagonal line, which is: x=2 the inverse of a line with points (x,y) is a line with points (y,x) for y=2, a point on the line is (0,2) on the inverse line x=2, the corresponding point is (2,0) x and y are switched.

Answer 2

Problem 1) No, not one-to-one. A one-to-one function must pass the horizontal line test (Draw a horizontal line and move it around anywhere on the graph. If it hits the graph at more than one spot anywhere, it it not one-to-one). If you drew the horizontal line y = 3, it would hit the graph at both x = -1 and x = 11.

Problem 2) If you meant to say g(x) = 4/x, then:

f(g(x)) = 4/(4/x) = (4/1)•(x/4) = x

g(f(x)) = 4/(4/x) = (4/1)•(x/4) = x

So, f and g are inverses