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Sagot :
### Part (a): Finding the Slant Asymptote
To find the slant asymptote of the rational function [tex]\( f(x) = \frac{x^2 - 64}{x} \)[/tex], we perform polynomial division.
Step-by-Step Polynomial Division:
1. Division Process:
- Divide the leading term of the numerator [tex]\( x^2 \)[/tex] by the leading term of the denominator [tex]\( x \)[/tex].
- [tex]\( x^2 \div x = x \)[/tex].
2. Multiply and Subtract:
- Multiply [tex]\( x \)[/tex] (the result from the previous step) by the denominator [tex]\( x \)[/tex] and subtract from the original polynomial [tex]\( x^2 - 64 \)[/tex].
- This gives: [tex]\( x^2 - 64 - (x \cdot x) = x^2 - 64 - x^2 = -64 \)[/tex].
3. Result of Division:
- The quotient from this division process is [tex]\( x \)[/tex].
- Since the degree of [tex]\( -64 \)[/tex] (the remainder) is lower than the degree of the denominator [tex]\( x \)[/tex], the division is complete.
- Hence, the slant asymptote is represented by the quotient term, which is [tex]\( y = x \)[/tex].
Thus, the correct choice is:
- A. The equation of the slant asymptote is [tex]\( y = x \)[/tex].
### Part (b): Determine the Symmetry of the Graph
To determine the symmetry of the graph of [tex]\( f(x) = \frac{x^2 - 64}{x} \)[/tex], we need to check for the following types of symmetry:
1. Y-axis Symmetry:
- For y-axis symmetry, [tex]\( f(-x) \)[/tex] should equal [tex]\( f(x) \)[/tex].
- Calculate: [tex]\( f(-x) = \frac{(-x)^2 - 64}{-x} = \frac{x^2 - 64}{-x} = -\frac{x^2 - 64}{x} = -f(x) \)[/tex].
- This shows [tex]\( f(-x) = -f(x) \)[/tex], which does not equal [tex]\( f(x) \)[/tex].
2. Origin Symmetry:
- For origin symmetry, [tex]\( f(-x) \)[/tex] should equal [tex]\(-f(x) \)[/tex].
- From the previous calculation, [tex]\( f(-x) = -f(x) \)[/tex].
- This condition is satisfied, implying origin symmetry.
Thus, the correct answer for symmetry determination is:
- Origin symmetry
### Summary of Choices
Part (a):
A. The equation of the slant asymptote is [tex]\( y = x \)[/tex].
Part (b):
- Origin symmetry
To find the slant asymptote of the rational function [tex]\( f(x) = \frac{x^2 - 64}{x} \)[/tex], we perform polynomial division.
Step-by-Step Polynomial Division:
1. Division Process:
- Divide the leading term of the numerator [tex]\( x^2 \)[/tex] by the leading term of the denominator [tex]\( x \)[/tex].
- [tex]\( x^2 \div x = x \)[/tex].
2. Multiply and Subtract:
- Multiply [tex]\( x \)[/tex] (the result from the previous step) by the denominator [tex]\( x \)[/tex] and subtract from the original polynomial [tex]\( x^2 - 64 \)[/tex].
- This gives: [tex]\( x^2 - 64 - (x \cdot x) = x^2 - 64 - x^2 = -64 \)[/tex].
3. Result of Division:
- The quotient from this division process is [tex]\( x \)[/tex].
- Since the degree of [tex]\( -64 \)[/tex] (the remainder) is lower than the degree of the denominator [tex]\( x \)[/tex], the division is complete.
- Hence, the slant asymptote is represented by the quotient term, which is [tex]\( y = x \)[/tex].
Thus, the correct choice is:
- A. The equation of the slant asymptote is [tex]\( y = x \)[/tex].
### Part (b): Determine the Symmetry of the Graph
To determine the symmetry of the graph of [tex]\( f(x) = \frac{x^2 - 64}{x} \)[/tex], we need to check for the following types of symmetry:
1. Y-axis Symmetry:
- For y-axis symmetry, [tex]\( f(-x) \)[/tex] should equal [tex]\( f(x) \)[/tex].
- Calculate: [tex]\( f(-x) = \frac{(-x)^2 - 64}{-x} = \frac{x^2 - 64}{-x} = -\frac{x^2 - 64}{x} = -f(x) \)[/tex].
- This shows [tex]\( f(-x) = -f(x) \)[/tex], which does not equal [tex]\( f(x) \)[/tex].
2. Origin Symmetry:
- For origin symmetry, [tex]\( f(-x) \)[/tex] should equal [tex]\(-f(x) \)[/tex].
- From the previous calculation, [tex]\( f(-x) = -f(x) \)[/tex].
- This condition is satisfied, implying origin symmetry.
Thus, the correct answer for symmetry determination is:
- Origin symmetry
### Summary of Choices
Part (a):
A. The equation of the slant asymptote is [tex]\( y = x \)[/tex].
Part (b):
- Origin symmetry
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