QuestionFind the domain, range, inverse domain, and inverse range of the function $f(x)=2x-6$ .

Studdy Solution

STEP 1

Assumptions1. The function is given as $f(x)=x-6$ . We are looking for the domain and range of the function and its inverse.

STEP 2

The domain of a function is the set of all possible input values (usually x-values) which will produce a valid output from a particular function. For the given function $f(x)=2x-6$ , there are no restrictions on the input values. Therefore, the domain is all real numbers.
$Domain\, of\, f(x) = (-\infty, \infty)$

STEP 3

The range of a function is the set of all possible output values (usually y-values), which result from the function. For the given function $f(x)=2x-6$ , there are no restrictions on the output values. Therefore, the range is also all real numbers.
$Range\, of\, f(x) = (-\infty, \infty)$

STEP 4

To find the domain and range of the inverse function, we first need to find the inverse function. The inverse of a function 'undoes' the operation of the original function. In other words, if $f(x) = y$ , then $f^{-1}(y) = x$ .To find the inverse of the function $f(x)=2x-6$ , we first replace $f(x)$ with $y$ .
$y =2x -6$

STEP 5

Next, we swap $x$ and $y$ to find the inverse function.
$x =2y -$

STEP 6

Now, solve for $y$ to get the inverse function.
$y = \frac{x+6}{2}$ So, the inverse function is $f^{-1}(x) = \frac{x+6}{2}$ .

STEP 7

The domain of the inverse function is the set of all possible input values (usually x-values) which will produce a valid output from the inverse function. For the given inverse function $f^{-1}(x) = \frac{x+6}{2}$ , there are no restrictions on the input values. Therefore, the domain is all real numbers.
$Domain\, of\, f^{-1}(x) = (-\infty, \infty)$

STEP 8

The range of the inverse function is the set of all possible output values (usually y-values), which result from the inverse function. For the given inverse function $f^{-1}(x) = \frac{x+6}{2}$ , there are no restrictions on the output values. Therefore, the range is also all real numbers.
$Range\, of\, f^{-1}(x) = (-\infty, \infty)$ So, the domain and range of the function $f(x)=2x-6$ and its inverse $f^{-1}(x) = \frac{x+6}{2}$ are all real numbers.

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