Answer :
To determine whether the statement "If the equation of a function is a rational expression, the function is rational" is true or false, we need to understand the definition of a rational function and a rational expression.
### Definitions
1. Rational Expression: A rational expression is a ratio (or fraction) of two polynomials. Formally, an expression of the form [tex]\( \frac{P(x)}{Q(x)} \)[/tex], where [tex]\( P(x) \)[/tex] and [tex]\( Q(x) \)[/tex] are polynomials, and [tex]\( Q(x) \neq 0 \)[/tex].
2. Rational Function: A rational function is a function that can be expressed as a ratio of two polynomials. Formally, [tex]\( f(x) = \frac{P(x)}{Q(x)} \)[/tex], where [tex]\( P(x) \)[/tex] and [tex]\( Q(x) \)[/tex] are polynomials, and [tex]\( Q(x) \neq 0 \)[/tex].
### Explanation
- A rational function, by definition, is any function that can be expressed as the ratio of two polynomials.
- A rational expression is also defined as a ratio of two polynomials.
Since a rational function and a rational expression share the same requirement—they are both ratios of polynomials—it follows that if the equation of a function is a rational expression, then by definition, the function itself is a rational function.
### Conclusion
Given the definitions and the logical equivalence between rational expressions and rational functions, the statement is indeed true.
Answer:
A. True
### Definitions
1. Rational Expression: A rational expression is a ratio (or fraction) of two polynomials. Formally, an expression of the form [tex]\( \frac{P(x)}{Q(x)} \)[/tex], where [tex]\( P(x) \)[/tex] and [tex]\( Q(x) \)[/tex] are polynomials, and [tex]\( Q(x) \neq 0 \)[/tex].
2. Rational Function: A rational function is a function that can be expressed as a ratio of two polynomials. Formally, [tex]\( f(x) = \frac{P(x)}{Q(x)} \)[/tex], where [tex]\( P(x) \)[/tex] and [tex]\( Q(x) \)[/tex] are polynomials, and [tex]\( Q(x) \neq 0 \)[/tex].
### Explanation
- A rational function, by definition, is any function that can be expressed as the ratio of two polynomials.
- A rational expression is also defined as a ratio of two polynomials.
Since a rational function and a rational expression share the same requirement—they are both ratios of polynomials—it follows that if the equation of a function is a rational expression, then by definition, the function itself is a rational function.
### Conclusion
Given the definitions and the logical equivalence between rational expressions and rational functions, the statement is indeed true.
Answer:
A. True
