Question

Consider . Describe and provide an explanation for the end behavior of j(x).

Answer 1

It seems you're asking about the end behavior of a function \( j(x) \), but the specific details about the function are missing. To analyze the end behavior of a function, we would typically look at its leading term if it's a polynomial, or consider limits if it's a rational function or other types.

Here's how you can generally describe the end behavior based on different types of functions:

1. **Polynomial Functions**:
- For a polynomial \( j(x) = ax^n + bx^{n-1} + ... + k \):
- If \( a > 0 \) and \( n \) is even, \( j(x) \to +\infty \) as \( x \to +\infty \) and \( j(x) \to +\infty \) as \( x \to -\infty \).
- If \( a > 0 \) and \( n \) is odd, \( j(x) \to +\infty \) as \( x \to +\infty \) and \( j(x) \to -\infty \) as \( x \to -\infty \).
- If \( a < 0 \) and \( n \) is even, \( j(x) \to -\infty \) as \( x \to +\infty \) and \( j(x) \to -\infty \) as \( x \to -\infty \).
- If \( a < 0 \) and \( n \) is odd, \( j(x) \to -\infty \) as \( x \to +\infty \) and \( j(x) \to +\infty \) as \( x \to -\infty \).

2. **Rational Functions**:
- For a rational function \( j(x) = \frac{p(x)}{q(x)} \):
- Analyze the degrees of the polynomial in the numerator \( p(x) \) and the denominator \( q(x) \).
- If the degree of \( p(x) \) is less than the degree of \( q(x) \), \( j(x) \to 0 \) as \( x \to \pm\infty \).
- If the degree of \( p(x) \) is equal to the degree of \( q(x) \), then \( j(x) \to \frac{a}{b} \), where \( a \) and \( b \) are the leading coefficients of \( p(x) \) and \( q(x) \) respectively.
- If the degree of \( p(x) \) is greater than the degree of \( q(x) \), \( j(x) \to \pm \infty \) depending on the sign of the leading coefficients.

3. **Exponential Functions**:
- For exponential functions \( j(x) = a \cdot b^x \):
- If \( b > 1 \), \( j(x) \to \infty \) as \( x \to \infty \) and \( j(x) \to 0 \) as \( x \to -\infty \).
- If \( 0 < b < 1 \), \( j(x) \to 0 \) as \( x \to \infty \) and \( j(x) \to \infty \) as \( x \to -\infty \).

4. **Logarithmic Functions**:
- For logarithmic functions like \( j(x) = \log_b(x) \):
- \( j(x) \to -\infty \) as \( x \to 0^+ \) and \( j(x) \to \infty \) as \( x \to \infty \).

If you can provide more information about the specific function \( j(x) \), I can offer a more tailored analysis of its end behavior.