Question

Sketch the graph of the given function. Then state the function’s domain and range. f(x)= 1/2(5)^x+3

Answer 1

Graph of the following function is attached with the answer.

Domain : ( - ∞ , + ∞ )

Range : ( 3 , + ∞ )

Explanation:

`f(x)=(1)/(2)x^(2)+3`

Domain of any quadratic equation is from negative infinity to positive infinity under no restrictions.

So, Domain : ( - ∞ , + ∞ )

The Range of any function can be calculated easily if there is just one term with variable. The method to find Range by that method is explained with the example as follows:

1. Range of x : ( - ∞ , + ∞ )
2. Range of
   `\textrm{x}^(2)` : [ 0 , + ∞ ) as every square number is more than or equal to zero.
3. Range of
   `(1)/(2)\textrm{x}^(2)` : [ 0 , + ∞ ) as 0/2 = 0 and ∞/2 = ∞.
4. Range of
   `(1)/(2)\textrm{x}^(2)+3` : [ 3 , + ∞ ) as 0 + 3 = 3 and ∞ + 3 = ∞.

Therefore the Range of
`\mathbf{f(x)\boldsymbol=(1)/(2)x^(2)\boldsymbol+3}` is [ 3 , + ∞ )

(NOTE : [a,b] means all the numbers between 'a' and 'b' including 'a' and 'b'.

(a,b) means all the numbers between 'a' and 'b' excluding 'a' and 'b'.

(a,b] means all the numbers between 'a' and 'b' including only 'b' not 'a'.

[a,b) means all the numbers between 'a' and 'b' including only 'a' not 'b'.

{a,b} means only 'a' and 'b'.

{a,b] or (a,b} doesn't mean anything. )