Find the domain and range of f(x)=x2+x−36
.

Syntax Note:

Type R for all real numbers
The interval (2,5)
 would be written as 2
 and x
 
The interval [−1,2)
 would be written as −1
 and x
 
The interval (−∞,3)
 would be written as x
The interval [7,∞)
 would be written as x>=7


Domain = 

 
Range =

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.

Find the domain and range of f(x)=x^2+x−36

Learn how to find the domain and range of the quadratic function f(x) = x^2 + x − 36. This detailed solution covers key algebraic concepts, including finding the vertex and analyzing the function's behavior. Ideal for students in Grades 9-11.

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