Determine whether the statements are True or False. Justify your answer with an explanation. If f(x) = x + 1 and g(x) = 6x, then (f ○ g)(x) = (g ○ f)(x).

If you are given two functions f(x) and g(x), you can calculate (f ○ g)(x) if and only if the range of g is a subset of the domain of f.

Problem 1

Explanation:

The notation (f o g)(x) means f( g(x) ). Here g(x) is the inner function.

So,

f(x) = x+1

f( g(x) ) = g(x) + 1 replace every x with g(x)

f( g(x) ) = 6x+1 ... plug in g(x) = 6x

(f o g)(x) = 6x+1

Now let's flip things around

g(x) = 6x

g( f(x) ) = 6*( f(x) ) replace every x with f(x)

g( f(x) ) = 6(x+1) plug in f(x) = x+1

g( f(x) ) = 6x+6

(g o f)(x) = 6x+6

This shows that (f o g)(x) = (g o f)(x)  is a false equation for the given f(x) and g(x) functions.

Problem 2

Explanation:

Let's say that g(x) produced a number that wasn't in the domain of f(x). This would mean that f( g(x) ) would be undefined.

For example, let

f(x) = 1/(x+2)

g(x) = -2

The g(x) function will always produce the output -2 regardless of what the input x is. Feeding that -2 output into f(x) leads to 1/(x+2) = 1/(-2+2) = 1/0 which is undefined.

So it's important that the outputs of g(x) line up with the domain of f(x). Outputs of g(x) must be valid inputs of f(x).

the one where 1 and 2 are directly across from each other.  

see image

answer-

a. the total cost of a taxi ride includes an initial charge of approximately $3.30

solution-

in the graph,

x axis represents the distance traveled and

y axis represents the total fare

the best fit line intersects the y axis approximately at 3.30 or y-intercept is 3.30. at x=0, i.e when the taxi is still, the value of y or the cost of the fare is 3.30. that means $3.30 is the fixed charge which will be levied along with the other fare.

the slope of the line represent the cost per mile. as the best fit line is straight line, so the price per mile is constant.