I am so tired and confused please help me with my Math Homework quickly please

Question

1.)

Use the horizontal line test to identify the relation whose inverse is a function.

A.)

B.)

C.)

D.)

2.) Identify the inverse of f(x) = 4 + x2. Determine whether it is a function and state its domain and range.

A.)

B.)

C.)

D.)

Please help

Answer 1

1) The problem is asking you which of these graphs will have an inverse that is a function. The inverse relation will be a function if the original graph passes the horizontal line test. So if you can draw a horizontal line that crosses the graph in more than one place, then the inverse will NOT be a function. If all horizontal lines cross no more than once, the inverse will be a function. For example, look at graph B). If you draw the horizontal line y = -2, the line will cross the graph five times. So the inverse of B) is NOT a function.

2) f(x) = x² + 4 is a quadratic equation, so its graph is a parabola. A parabola fails the horizontal line test, so the inverse of f(x) will not be a function. To find the inverse:

y = x² + 4

Change the x to y and the y to x, then solve for y:

x = y² + 4

x - 4 = y²

±√(x-4) = y

±√(x-4) = f^-1(x)

We put the domain restriction that x ≥ 4, because you can't take the square root of a negative number in the Real Number set.