Same Base, But Different Exponents

Algebra Level 3

Let $f(x)=\left ( 1-x^3 \right )^{2+\sqrt{5}}$ , $\;g(x)=\left ( 1-x^3 \right )^{2-\sqrt{5}}$ and $h(x)=\left ( 1-x^3 \right )^{-(2+\sqrt{5})}$ .

Which of these affirmations about their real domains is true ?

The domains of

f(x)

and

g(x)

are the same The domains of

f(x)

and

h(x)

are the same The domains of

f(x)

g(x)

and

h(x)

are all the same The domains of

g(x)

and

h(x)

are the same The domains of

f(x)

g(x)

and

h(x)

are different from each other

1 solution

Jaydee Lucero
Jun 27, 2017

Note that $2+\sqrt{5}$ is positive, while $2-\sqrt{5}$ and $-(2+\sqrt{5})$ are both negative. So write

$f(x)=(1-x^3)^{2+\sqrt{5}},g(x)=\frac{1}{(1-x^3)^{-2+\sqrt{5}}},h(x)=\frac{1}{(1-x^3)^{2+\sqrt{5}}}$

Note that $1-x^3 = (1-x)(1+x+x^2)$ . Therefore, the domain of $f$ is all real numbers, while the domain of $g$ and $h$ are both all real numbers except $x=1$ . Thus, the domains of $g$ and $h$ are the same .

Same Base, But Different Exponents

1 solution

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