Question

F(x) = x

f(x) = x^2
f(x) = 2^x
1) Which functions intersect?

2) How many points of intersection are there?

3) What does a point of intersection mean?

Answer 1

Part 1) In the procedure

Part 2) There are 4 points of intersection (see the procedure)

Part 3) The answer in the procedure

Explanation:

we have

`f(x)=x` -----> linear function

`f(x)=x^(2)` ----> quadratic function

`f(x)=2^(x)` ----> exponential function

Part 1) Which functions intersect?

using a graphing tool

we have that

`f(x)=x` intersect with
`f(x)=x^(2)`

`f(x)=x^(2)` intersect with
`f(x)=x` and
`f(x)=2^(x)`

`f(x)=2^(x)` intersect with
`f(x)=x^(2)`

see the attached figure

Part 2) How many points of intersection are there?

In total there are 4 points of intersection

so

Between

`f(x)=x` and
`f(x)=x^(2)`

there are 2 points -----> (0,0) and (1,1)

Between

`f(x)=2^(x)` and
`f(x)=x^(2)`

there are 2 points -----> (-0.77,0.59) and (2,4)

Part 3) What does a point of intersection mean?

we know that

A point of intersection between two functions means a common solution for both functions.

so

Example 1

The point (0,0) is a point of intersection between
`f(x)=x` and
`f(x)=x^(2)`

For x=0

Find the value of both functions

`f(x)=x` ----->
`f(0)=0`

`f(x)=x^(2)` ---->
`f(0)=0^(2)=0`

Both functions have the same value

Example 2

The point (2,4) is a point of intersection between
`f(x)=2^(x)` and
`f(x)=x^(2)`

For x=2

Find the value of both functions

`f(x)=2^(x)` ----->
`f(2)=2^(2)=4`

`f(x)=x^(2)` ---->
`f(2)=2^(2)=4`

Both functions have the same value