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larisa [96]
3 years ago
11

Which of these functions has an inverse function? Select all that apply.

Mathematics
2 answers:
allochka39001 [22]3 years ago
6 0

Answer:

The function x=y and y=x^{3} are inverse function.

Step-by-step explanation:

Function inverse definition:

If a provided function <em>f(x)</em> is mapped <em>x</em> to <em>y</em>, then the inverse of the provided function <em>f(x) </em>is mapped <em>y</em> to <em>x</em>.

Or  f(x)=f(y)\Rightarrow x=y

Now, consider the function <em>y</em> =<em> x</em>.

Interchange the variables <em>x</em> and <em>y</em>.

x=y

Now, solve x=y for y.

x=y

Therefore, this function has an inverse.

Consider the function y=x^{2}.

Interchange the variables <em>x</em> and <em>y</em>.

x=y^{2}

Now, solve x=y^{2} for y.

\pm \sqrt{x} =y

Therefore, the function has no inverse.

Consider the function y=x^{3}.

Interchange the variables <em>x</em> and <em>y</em>.

x=y^{3}

Now, solve x=y^{3} for y.

\sqrt[3]{x}=y

Therefore, the function has inverse.

Consider the function y=x^{4}.

Interchange the variables <em>x</em> and <em>y</em>.

x=y^{4}

Now, solve x=y^{4} for y.

\pm \sqrt[4]{x}=y

Therefore, the function has no inverse.

Hence, the function x=y and y=x^{3} are inverse function.

ElenaW [278]3 years ago
5 0

ANSWER

y=x

y=x³

EXPLANATION

A function, f(x) has an inverse if and only if

f(a) = f(b)   \Rightarrow \: a = b

Thus, the function is one to one.

For y=x or

f(x) = x

f(a) = f(b)   \Rightarrow \: a = b

Hence this function has an inverse.

For the function y=x² or f(x)=x².

f(a) = f(b)   \Rightarrow \:  {a}^{2} =  {b}^{2} \Rightarrow \: a =  \pm \: b

This function has no inverse on the entire real numbers.

For the function y=x³ or f(x)=x³

f(a) = f(b)   \Rightarrow \:  {a}^{3} =  {b}^{3} \Rightarrow \: a = b

This function also has an inverse.

For y=x⁴ or f(x) =x⁴

f(a) = f(b)   \Rightarrow \:  {a}^{4} =  {b}^{4} \Rightarrow \: a =  \pm \: b

This function has no inverse over the entire real numbers.

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