What is the range of f(x)=(x+4)/(x^2+x-3) ?

The range of f(x)=(x+4)/(x^2+x-3) is f(x) =-1/13

English

Español

Português

Français

Deutsch

Italiano

Русский

中文(简体)

한국어

日本語

Tiếng Việt

עברית

العربية

Popular >

range of f(x)=(x+4)/(x^2+x-3)

range

f (x) = \frac{x + 4}{x ^{2} + x - 3}

f (x) \leq - 1 or f (x) \geq - \frac{1}{13}

Interval Notation

(- \infty, - 1] \cup [- \frac{1}{13}, \infty)

Solution steps

Rewrite as

\frac{x + 4}{x ^{2} + x - 3} = y

Multiply both sides by

x^{2} + x - 3

x + 4 = y (x^{2} + x - 3)

The range is the set of y for which the discriminant is greater or equal to zero

Discriminant

x + 4 = y (x^{2} + x - 3) : 13 y^{2} + 14 y + 1

13 y^{2} + 14 y + 1 \geq 0 : y \leq - 1 or y \geq - \frac{1}{13}

Check if the range interval endpoints are included

y = - 1 :

Included

y = - \frac{1}{13} :

Included

Therefore the range is

f (x) \leq - 1 or f (x) \geq - \frac{1}{13}

2 (\frac{1}{3})^{x}

f (x) = 5 x^{2} + 20 x - 17

\frac{x - 3}{x ^{2} - 1}

f (x) = e^{x - 2}

f (x) = - \frac{3}{x}

What is the range of f(x)=(x+4)/(x^2+x-3) ?
The range of f(x)=(x+4)/(x^2+x-3) is f(x)<=-1\lor f(x)>=-1/13