text: # Conditions of exponential functions
  
  Okay, is the above statement correct? Because if I put $$x=\frac{1}{2}$$ then $$f(x)$$ will have two values. So will that remain a function anymore?
  
  • Working with exponential functions (as defined above), we only consider the principal root. For example, $a^{\frac{11}{4}}$ could be interpreted as $\sqrt[4]{a^{11}}$. This is perfect for defining a function. But you can't use this notion everywhere. Consider $(-3)^2=9$, which means $(-3)^{2\cdot\frac{1}{2}}=9^{\frac{1}{2}}$, so $(-3)^1=-3=3$? This is wrong of course. – Corellian Oct 19 '18 at 5:39
  
  Indeed, the unrestricted inverse of any non-injective function is multivalued, but this is fixed by simply restricting the co-domain to some interval. Take $$f(x)=y=x^2$$ for example. It's inverse satisfies $$f^{-1}(y) = x$$. Therefore, if we just solve our original function for $$x$$, we should have our answer.
  $$x^2 = y \\ x^2 - y = 0 \\ (x+y^{\frac{1}{2}})(x-y^{\frac{1}{2}})=0 \\ f^{-1}(y) = x = \pm y^{\frac{1}{2}}$$
  Clearly, this is multivalued, and cannot be a function, so by convention, we define the principal square root function to be positive solution to be the inverse of $$f(x) = y = x^2$$.
  In conclusion, the statement is correct because $$f(x)=a^x$$ only has one value at $$f(\frac{1}{2})$$.
explanation: The text extract discusses the conditions of exponential functions and addresses a concern about whether a function can have multiple values for a single input, specifically when \( x = \frac{1}{2} \). The user raises a question about the correctness of the statement and provides an example to illustrate their concern. The response explains that the principal root is considered in exponential functions to ensure they remain single-valued and provides a detailed explanation using the example of the square root function.
distilabel_metadata.content: Evaluate the user given text extract and determine if it contains a math question and optionally and answer to this question.
  
  Output rules:
  
  1. Briefly describe the document and mention if it contains a question or a question and answer.
  2. Finalize the output with: Label: no-question,question,question-and-answer,multiple-question,multiple-question-and-answer
distilabel_metadata.role: system
distilabel_metadata.content: # Conditions of exponential functions
  
  Okay, is the above statement correct? Because if I put $$x=\frac{1}{2}$$ then $$f(x)$$ will have two values. So will that remain a function anymore?
  
  • Working with exponential functions (as defined above), we only consider the principal root. For example, $a^{\frac{11}{4}}$ could be interpreted as $\sqrt[4]{a^{11}}$. This is perfect for defining a function. But you can't use this notion everywhere. Consider $(-3)^2=9$, which means $(-3)^{2\cdot\frac{1}{2}}=9^{\frac{1}{2}}$, so $(-3)^1=-3=3$? This is wrong of course. – Corellian Oct 19 '18 at 5:39
  
  Indeed, the unrestricted inverse of any non-injective function is multivalued, but this is fixed by simply restricting the co-domain to some interval. Take $$f(x)=y=x^2$$ for example. It's inverse satisfies $$f^{-1}(y) = x$$. Therefore, if we just solve our original function for $$x$$, we should have our answer.
  $$x^2 = y \\ x^2 - y = 0 \\ (x+y^{\frac{1}{2}})(x-y^{\frac{1}{2}})=0 \\ f^{-1}(y) = x = \pm y^{\frac{1}{2}}$$
  Clearly, this is multivalued, and cannot be a function, so by convention, we define the principal square root function to be positive solution to be the inverse of $$f(x) = y = x^2$$.
  In conclusion, the statement is correct because $$f(x)=a^x$$ only has one value at $$f(\frac{1}{2})$$.
distilabel_metadata.role: user
distilabel_metadata.raw_output_q_a_detector_0: The text extract discusses the conditions of exponential functions and addresses a concern about whether a function can have multiple values for a single input, specifically when \( x = \frac{1}{2} \). The user raises a question about the correctness of the statement and provides an example to illustrate their concern. The response explains that the principal root is considered in exponential functions to ensure they remain single-valued and provides a detailed explanation using the example of the square root function.
  
  Label: question-and-answer