Question

Find the value of h(45) for the function below. h(x)=1/9(90 - x). A. 15 B. 5 C. -35 D. -315

Answer 1

Answer:

Inverse Functions

One-to-one

Suppose f : A ⇥ B is a function. We call f one-to-one if every distinct

pair of objects in A is assigned to a distinct pair of objects in B. In other

words, each object of the target has at most one object from the domain

assigned to it.

There is a way of phrasing the previous definition in a more mathematical

language: f is one-to-one if whenever we have two objects a, c ⇤ A with

a ⌅= c, we are guaranteed that f(a) ⌅= f(c).

Example. f : R ⇥ R where f(x) = x2 is not one-to-one because 3 ⌅= 3

and yet f(3) = f(3) since f(3) and f(3) both equal 9.

Horizontal line test

If a horizontal line intersects the graph of f(x) in more than one point,

then f(x) is not one-to-one.

The reason f(x) would not be one-to-one is that the graph would contain

two points that have the same second coordinate – for example, (2, 3) and

(4, 3). That would mean that f(2) and f(4) both equal 3, and one-to-one

functions can’t assign two dierent objects in the domain to the same object

of the target.

If every horizontal line in R2 intersects the graph of a function at most

once, then the function is one-to-one.

Examples. Below is the graph of f : R ⇥ R where f(x) = x2. There is a

horizontal line that intersects this graph in more than one point, so f is not

one-to-one.

66

Inverse Functions

One-to-one

Suppose f : A —* B is a function. We call f one-to-one if every distinct

pair of objects in A is assigned to a distinct pair of objects in B. In other

words, each object of the target has at most one object from the domain

assigned to it.

There is a way of phrasing the previous definition in a more mathematical

language: f is one-to-one if whenever we have two objects a, c e A with

a ~ c, we are guaranteed that f(a) $ f(c).

Example. f : IR —* JR where f(x) = x2 is not one-to-one because 3 ~ —3

and yet f(3) = f(—3) since f(3) and f(—3) both equal 9.

Horizontal line test

If a horizontal line intersects the graph of f(.x) in more than one point,

then f(z) is not one-to-one.

The reason f(x) would not be one-to-one is that the graph would contain

two points that have the same second coordinate — for example, (2,3) and

(4,3). That would mean that f(2) and f(4) both equal 3, and one-to-one

functions can’t assign two different objects in the domain to the same object

of the target.

If every horizontal line in JR2 intersects the graph of a function at most

once, then the function is one-to-one.

Examples. Below is the graph of f : JR —, R where f(z) = z2. There is a

horizontal line that intersects this h in more than one point, so f is not

Step-by-step explanation: