For the function $f(x) = x^2 + x - 36$, let's find the domain and range.

1. Domain:

Since $f(x)$ is a quadratic function, it is defined for all real numbers. There are no restrictions like division by zero or taking the square root of a negative number.

• Domain: $R$ (all real numbers).

2. Range:

A quadratic function opens upwards or downwards depending on the coefficient of $x^2$. Here, the coefficient of $x^2$ is positive (1), so the parabola opens upwards. The vertex of the parabola will give us the minimum value, which determines the range.

Vertex:

To find the vertex, we use the formula for the x-coordinate of the vertex: $x = \frac{-b}{2a}$ For $f(x) = x^2 + x - 36$, the coefficients are:

• $a = 1$
• $b = 1$

Substitute into the vertex formula: $x = \frac{-1}{2(1)} = -\frac{1}{2}$

Now, substitute $x = -\frac{1}{2}$ back into the function to find the minimum value: $f\left(-\frac{1}{2}\right) = \left(-\frac{1}{2}\right)^2 + \left(-\frac{1}{2}\right) - 36 = \frac{1}{4} - \frac{1}{2} - 36 = -\frac{143}{4}$

Thus, the minimum value of the function is $-\frac{143}{4}$, and since the parabola opens upwards, the range is all values greater than or equal to this minimum.

• Range: $f(x) \geq -\frac{143}{4}$

Final Answer:

• Domain: $R$
• Range: $f(x) \geq -\frac{143}{4}$ or $x >= -\frac{143}{4}$

Would you like more details or have any questions?

5 Related Questions:

1. How do you determine whether a parabola opens upwards or downwards?
2. Can you explain how to complete the square for quadratic functions?
3. What is the significance of the vertex in the graph of a quadratic function?
4. How can you find the x-intercepts and y-intercepts of a quadratic function?
5. What is the difference between domain and range for different types of functions?

Tip: Always analyze the behavior of the quadratic function by focusing on the vertex and whether the parabola opens up or down. This helps in determining the range quickly.