For what real values of x is the rational expression g(x) = \frac{x^3 - 3x^2}{\frac{2}{x-2… — ACT

For what real values of $x$ is the rational expression $g(x) = \frac{x^3 - 3x^2}{\frac{2}{x-2} - \frac{x+4}{x^2 - 4}}$ undefined? Determine these three values and enter them in the blanks below in order from least to greatest.

Cevap:The values of

x

for which

g(x)

is undefined, ordered from least to greatest, are

x =

【-2】,

x =

【0】, and

x =

【2】.

Cevap

The values of

x

for which the expression is undefined, ordered from least to greatest, are

-2

0

, and

2

The rational expression is undefined where any constituent denominator is equal to zero, or where the entire main denominator is equal to zero. The constituent denominators are

x-2

and

x^2-4

, which are zero when

x = 2

x = -2

. The entire main denominator simplifies to

\frac{x}{(x-2)(x+2)}

, which is zero when

x = 0

. Therefore, the three values of

x

that make the expression undefined are

-2

0

, and

2

Adım Adım Çözüm

Identify the values of

x

that make the denominators of the individual rational terms equal to zero.

The denominators are

x - 2

and

x^2 - 4

. Setting these to zero gives

x = 2

and

x = \pm 2

. Thus,

x = 2

and

x = -2

are values that make the individual terms undefined.

Any rational term is undefined if its denominator is equal to zero.

Find the values of

x

that make the overall denominator of the main fraction equal to zero by first simplifying it.

The main denominator is

\frac{2}{x-2} - \frac{x+4}{x^2-4}

. Finding a common denominator yields

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

. Setting this equal to zero gives

x = 0

The main rational expression is undefined if its entire denominator is equal to zero.

Combine all restricted values and list them in order from least to greatest.

The restricted values are

x = -2

x = 0

, and

x = 2

These are all the distinct real numbers that cause any division by zero in the original expression.

Anahtar Kavram

Finding the domain of nested rational expressions

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