Certainly! Let's solve the given function step-by-step.
We are tasked with simplifying or finding the function [tex]\( t \)[/tex] which is defined as follows:
[tex]\[ t = \frac{2 x^2 - 1}{3 x + 5} \][/tex]
To understand this expression, let's break it down:
1. Numerator Analysis:
[tex]\[
2 x^2 - 1
\][/tex]
- This is a quadratic expression.
2. Denominator Analysis:
[tex]\[
3 x + 5
\][/tex]
- This is a linear expression in terms of [tex]\( x \)[/tex].
Next, let’s consider a few important aspects of the function:
- Domain Consideration:
To find the domain of the function, we need to ensure that the denominator is not zero since division by zero is undefined. Therefore, we solve for:
[tex]\[
3 x + 5 \neq 0
\][/tex]
Subtracting 5 from both sides we get:
[tex]\[
3 x \neq -5
\][/tex]
Dividing both sides by 3, we get:
[tex]\[
x \neq -\frac{5}{3}
\][/tex]
Hence, the function [tex]\( t \)[/tex] is defined for all real numbers [tex]\( x \)[/tex] except [tex]\( x = -\frac{5}{3} \)[/tex].
- Critical Points and Behavior:
The quadratic numerator [tex]\( 2 x^2 - 1 \)[/tex] and the linear denominator [tex]\( 3 x + 5 \)[/tex] create interesting behavior in the graph of the function [tex]\( t \)[/tex]. Key points of interests could be the roots of the numerator and points where [tex]\( t = 0 \)[/tex].
- Horizontal Asymptote:
To find the horizontal asymptote for this function, we compare the degrees of the polynomial in the numerator and the denominator. Since the degree of the numerator ([tex]\(2 x^2\)[/tex]) is one higher than the degree of the denominator ([tex]\(3 x\)[/tex]), the function does not have a horizontal asymptote. Instead, it has an oblique (slant) asymptote which occurs when the degree of the numerator is exactly one more than that of the denominator.
Finally, the function can be expressed as:
[tex]\[ t = \frac{2 x^2 - 1}{3 x + 5} \][/tex]
In summary:
- The function simplifies directly to the form provided.
- The domain is all real numbers except [tex]\( x = -\frac{5}{3} \)[/tex].
- The function does not have a horizontal asymptote but might have a slant asymptote to be determined by polynomial long division if needed.
- The behavior of the function should be analyzed by looking at critical points and roots.
