Question

Math: One to One function...help!

Answer 1

A one-to-one function is a type of function where each element in the domain is paired with a unique element in the codomain. It can be determined using the horizontal line test.

Step-by-step explanation:

A one-to-one function, also known as an injective function, is a type of function where each element in the domain is paired with a unique element in the codomain. In other words, no two different elements in the domain can have the same image in the codomain. To determine if a function is one-to-one, you can use the horizontal line test. If a horizontal line intersects the graph of the function at most once, then the function is one-to-one.

For example, the function f(x) = x^2 is not one-to-one because for every positive value of x, there are two different values of f(x) (x^2 and (-x)^2). On the other hand, the function f(x) = 2x + 3 is one-to-one because for every unique value of x, there is a unique value of f(x).

Answer 2

A one-to-one function has an inverse. The inverse is another function that undoes the action of the first one, so if we evaluate a function
`f` at some point
`x` to get the number
`f(x)`, evaluating the inverse at
`f(x)` will recover the original input
`x`. In other words,

`f^(-1)(f(x)) = x`

The process works in the opposite direction, too:

`f\left(f^(-1)(x)\right) = x`

From the given definition of
`g`, we have
`g(-4) = 3`, so taking inverses on both sides, we find

`g(-4) = 3 \implies g^(-1)(g(-4)) = g^(-1)(3) \implies \boxed{g^(-1)(3) = -4}`

Given
`h(x)=2x-13`, evaluating
`h` at its inverse will recover
`x`, so that

`h\left(h^(-1)(x)\right) = x \implies 2h^(-1)(x) - 13 = x \implies \boxed{h^(-1)(x) = \frac{x+13}2}`

`(h\circ h^(-1))(x)` is another way of writing the compound function
`h\left(h^(-1)(x)\right)`. As already discussed, this reduces to
`x`, so

`\boxed{\left(h\circ h^(-1)\right)(-9) = -9}`