1. **State the problem:** We are given the function $$y=\frac{4x-3}{2x^2+1}$$ and asked to analyze it. 2. **Recall the formula and rules:** This is a rational function where the numerator is $$4x-3$$ and the denominator is $$2x^2+1$$. 3. **Find domain:** The denominator $$2x^2+1$$ is always positive since $$2x^2 \geq 0$$ and adding 1 makes it strictly positive. So, the domain is all real numbers. 4. **Find intercepts:** - **y-intercept:** Set $$x=0$$: $$y=\frac{4(0)-3}{2(0)^2+1}=\frac{-3}{1}=-3$$ - **x-intercept:** Set $$y=0$$, so numerator must be zero: $$4x-3=0 \Rightarrow x=\frac{3}{4}$$ 5. **Find asymptotes:** - **Vertical asymptotes:** None, since denominator never zero. - **Horizontal asymptote:** For large $$x$$, degree numerator 1, denominator 2, so $$y\to 0$$. 6. **Simplify and analyze behavior:** No factorization possible to simplify. 7. **Summary:** - Domain: all real numbers - x-intercept: $$x=\frac{3}{4}$$ - y-intercept: $$y=-3$$ - Horizontal asymptote: $$y=0$$ - No vertical asymptotes Final answer: $$y=\frac{4x-3}{2x^2+1}$$ with the above properties.