Horizontal Asymptote: To find the horizontal asymptote, we look at the degrees of the polynomial in the numerator and the polynomial in the denominator. If the degree of the numerator is less than the degree of the denominator, the horizontal asymptote is $y = 0$ .
Vertical Asymptote: Since the numerator is a constant ( $5$ ) and the denominator is a linear term ( $2x+3$ ), the degree of the numerator ( $0$ ) is less than the degree of the denominator ( $1$ ). Therefore, the horizontal asymptote is $y = 0$ .
X-Intercept: To find the vertical asymptote, we set the denominator equal to $0$ and solve for $x$ . The vertical asymptote occurs where the function is undefined, which is where the denominator is $0$ .
Y-Intercept: Setting the denominator to zero gives us $2x + 3 = 0$ . Solving for $x$ , we get $x = -\frac{3}{2}$ . Therefore, the vertical asymptote is $x = -\frac{3}{2}$ .
No X-Intercept: To find the x-intercept, we set $f(x)$ to zero and solve for $x$ . The x-intercept occurs where the function crosses the x-axis, which is where $f(x) = 0$ .
No Hole: Setting $f(x)$ to zero gives us $0 = \frac{5}{2x+3}$ . Since the numerator is a non-zero constant, there is no value of $x$ that will make the function equal to zero. Therefore, there is no $x$ -intercept.
No Hole: Setting $f(x)$ to zero gives us $0 = \frac{5}{2x+3}$ . Since the numerator is a non-zero constant, there is no value of $x$ that will make the function equal to zero. Therefore, there is no $x$ -intercept.To find the $y$ -intercept, we set $x$ to zero and solve for $f(x)$ . The $y$ -intercept occurs where the function crosses the $y$ -axis, which is where $x = 0$ .
No Hole: Setting $f(x)$ to zero gives us $0 = \frac{5}{2x+3}$ . Since the numerator is a non-zero constant, there is no value of $x$ that will make the function equal to zero. Therefore, there is no $x$ -intercept.To find the $y$ -intercept, we set $x$ to zero and solve for $f(x)$ . The $y$ -intercept occurs where the function crosses the $y$ -axis, which is where $x = 0$ .Setting $x$ to zero gives us $0 = \frac{5}{2x+3}$ $1$ . Therefore, the $y$ -intercept is $0 = \frac{5}{2x+3}$ $3$ .
No Hole: Setting $f(x)$ to zero gives us $0 = \frac{5}{2x+3}$ . Since the numerator is a non-zero constant, there is no value of $x$ that will make the function equal to zero. Therefore, there is no $x$ -intercept.To find the $y$ -intercept, we set $x$ to zero and solve for $f(x)$ . The $y$ -intercept occurs where the function crosses the $y$ -axis, which is where $x = 0$ .Setting $x$ to zero gives us $0 = \frac{5}{2x+3}$ $1$ . Therefore, the $y$ -intercept is $0 = \frac{5}{2x+3}$ $3$ .To determine if there is a hole in the graph, we look for common factors in the numerator and denominator that can be canceled. If there is a common factor that is canceled, it indicates a hole at that $x$ -value.
No Hole: Setting $f(x)$ to zero gives us $0 = \frac{5}{2x+3}$ . Since the numerator is a non-zero constant, there is no value of $x$ that will make the function equal to zero. Therefore, there is no $x$ -intercept.To find the $y$ -intercept, we set $x$ to zero and solve for $f(x)$ . The $y$ -intercept occurs where the function crosses the $y$ -axis, which is where $x = 0$ .Setting $x$ to zero gives us $0 = \frac{5}{2x+3}$ $1$ . Therefore, the $y$ -intercept is $0 = \frac{5}{2x+3}$ $3$ .To determine if there is a hole in the graph, we look for common factors in the numerator and denominator that can be canceled. If there is a common factor that is canceled, it indicates a hole at that $x$ -value.Since the numerator is a constant and the denominator is a linear term without common factors, there is no cancellation possible. Therefore, there is no hole in the graph of the function.