Question

Let f(x)=e* and g(x)= X+6. What are the domain and range of (gºf)(x)?

Answer 1

Domain: Set of All Real Numbers. (-∞, ∞)

Range: (0, ∞)

Explanation:

`f(x)=e^(x)\\g(x)=x+6`

We need to evaluate (gof)(x). (gof)(x) means the composition of function g(x) and f(x). In order to find (gof)(x) replace every occurrence of x in g(x) with the expression of f(x) as shown below:

`(gof)(x)=g(f(x))\\\\ =g(e^(x))\\\\ =e^(x)+6`

We have to find the domain and range of this composite function.

Domain:

The expression
`e^(x)+6` is defined for all Real Numbers. It never gets undefined for any value of x. Hence the Domain is set of All Real Numbers.

Range:

The range of the parent exponential function
`e^(x)` is from 0 to positive infinity. The expression
`e^(x)+6` can be related to
`e^(x)` as a vertical shifted function by 6 units upward. As a result the range will also be shifted up. So the range of the composite function will be from 6 to positive infinity. In interval notation this would be expressed as: (0, ∞)