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QuestionFonction f(x)=2x−1x2+x−2f(x)=\frac{2 x-1}{x^{2}+x-2} : 1) Racines de x2+x−2x^{2}+x-2 ? 2) Ensemble de définition ? 3) lim⁡x→−∞f(x)\lim_{x \to -\infty} f(x) ? 4) f(−1)f(-1) ?

Studdy Solution

STEP 1

Assumptions1. The function is defined as f(x)=x−1x+x−f(x)=\frac{ x-1}{x^{}+x-} . We need to find the roots of the polynomial x+x−x^{}+x-
3. We need to find the domain of the function ff
4. We need to calculate the limit of the function as xx approaches −∞-\infty
5. We need to calculate the value of the function at x=−1x=-1

STEP 2

To find the roots of the polynomial x2+x−2x^{2}+x-2, we set the polynomial equal to zero and solve for xx.
x2+x−2=0x^{2}+x-2=0

STEP 3

This is a quadratic equation of the form ax2+bx+c=0ax^{2}+bx+c=0. We can solve it using the quadratic formula x=−b±b2−ac2ax=\frac{-b\pm\sqrt{b^{2}-ac}}{2a}.

STEP 4

Substitute the coefficients a=1a=1, b=1b=1, and c=−2c=-2 into the quadratic formula.
x=−1±12−4(1)(−2)2(1)x=\frac{-1\pm\sqrt{1^{2}-4(1)(-2)}}{2(1)}

STEP 5

implify the expression under the square root.
x=−1±1+82x=\frac{-1\pm\sqrt{1+8}}{2}

STEP 6

Calculate the square root.
x=−1±92x=\frac{-1\pm\sqrt{9}}{2}

STEP 7

implify the expression.
x=−1±32x=\frac{-1\pm3}{2}

STEP 8

Calculate the two possible values for xx.
x=−1+32=1,x=−1−32=−2x=\frac{-1+3}{2}=1, \quad x=\frac{-1-3}{2}=-2The roots of the polynomial x2+x−2x^{2}+x-2 are -2 and1.

STEP 9

The domain of the function ff is all real numbers except for the roots of the denominator polynomial x2+x−2x^{2}+x-2. We have already found the roots to be -2 and. Therefore, the domain of ff is R∖{−2,}R\setminus\{-2,\}.

STEP 10

To find the limit of the function as xx approaches −∞-\infty, we divide the numerator and denominator by x2x^{2}.
lim⁡x→−∞f(x)=lim⁡x→−∞2x/x2−/x2+x/x2−2/x2\lim{x \rightarrow-\infty} f(x)=\lim{x \rightarrow-\infty} \frac{2 x/x^{2}-/x^{2}}{+x/x^{2}-2/x^{2}}

STEP 11

implify the expression.
lim⁡x→−∞f(x)=lim⁡x→−∞/x−/x+/x−/x\lim{x \rightarrow-\infty} f(x)=\lim{x \rightarrow-\infty} \frac{/x-/x^{}}{+/x-/x^{}}

STEP 12

As xx approaches −∞-\infty, the terms 2/x2/x, /x2/x^{2}, /x/x, and 2/x22/x^{2} all approach0. Therefore, the limit is0.

STEP 13

To find the value of the function at x=−x=-, substitute x=−x=- into the function.
f(−)=2(−)−(−)2+(−)−2f(-)=\frac{2 (-)-}{(-)^{2}+(-)-2}

STEP 14

implify the expression.
f(−)=−2−−−2f(-)=\frac{-2-}{--2}

STEP 15

Calculate the value.
f(−)=−3−2=.5f(-)=\frac{-3}{-2}=.5The value of the function at x=−x=- is.5.

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