Question: "f(x)=−13(x+1)+1 Domain and range? Transformations? Name of Graph?"

High School
Geometry
High school geometry
Performing transformations

Trent Carpenter

Answered question

2021-03-18

$f (x) = \frac{- 1}{3 (x + 1)} + 1$
Domain and range?
Transformations?
Name of Graph?

Answer & Explanation

ensojadasH

Skilled2021-03-19Added 100 answers

Given the function $\frac{- 1}{3 (x + 1)} + 1$
We need to determine the domain and range of the function.
Domain is the set of input values for which the function is real and defined.
Since it is rational function the denominator should not be 0.
$3 x + 4 = 0$
$3 x = - 4$
$x = - \frac{4}{3}$
excluding $x = - \frac{4}{3}$ , all the values of x are supported
so $x ⟨ - \frac{4}{3} o r x ⟩ - \frac{4}{3}$
so the domain of the function is
$x \ln (\infty, - \frac{4}{3}) \cup (- \frac{4}{3}, \infty)$
Range is the set of values of the dependent variable for which a function is defined.
So first we need to find the inverse of the function.
$y = - \frac{1}{3 y + 4}$
swap x by y
$x (3 y + 4) = - 1$
$3 x y + 4 x = - 1$
$3 x y = - 1 - 4 x$
$y = \frac{- 1 - 4 x}{3 x}$
so the denominator should not be zero
$3 x = 0$
$x = 0$
sp excluding $x = 0$ all the values are supported.
so the range is $(- \infty, 0) \cup (0, \infty)$

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