Question

What are the domain and range of f(x)=log(x-1)+2?

Answer 1

Answer: Domain = $(-1, \infty)$

Range = $(-\infty, \infty)$

Step-by-step explanation:

Here, the given function,

$f(x) = log(x-1) + 2$

Since log( negative number ) = not defined

Also,

$log(0) = -\infty$

Hence, f(x) is defined if x - 1 > 0

⇒ x > 1

Thus, f(x) is defined for all the values that are greater than 1,

⇒ Domain of f(x) = All real number greater than 1,

= $(1, \infty)$

Now, for the interval $(1, \infty)$,

f(x) can be any real number.

Thus, the range of the given function = $(-\infty, \infty)$

Answer 2

Domain is set of values on which function is defined. Range is set of output values. The domain and range of given function is:

• Domain:  x > 1 , or $x \in$ (1, ∞ )
• Range:  x ∈ (-∞, ∞), $x \in \mathbb R$

What is domain and range of a function?

Domain is the set of values for which the given function is defined.

Range is the set of all values which the given function can output.

The given function is $f(x) = log(x-1) + 2$

Since we know that log function is defined for values > 0  as input.

Thus, x -1 > 0 or, x > 1

Thus, its domain is all values of x > 1

Since log function can output all values from -ve infinity to + infinity and x-1 can be from >0 to infinity, thus all input values are possible to log and thus, all output values of log are possible, and thus, the range of given function is all real numbers ( (-∞, ∞) + 2 = (-∞, ∞), thus, adding 2 won't change range )

And therefore, the domain and range of given function is obtained as:

• Domain:  x > 1 , or $x \in$ (1, ∞ )
• Range:  $x \in \mathbb R$

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