Find the range of `f(x) =x^(2) -5x + 6`

Answer

To find the range of the function \( f(x) = x^2 - 5x + 6 \), we can follow these steps:

### Step 1: Rewrite the function
Let \( y = f(x) = x^2 - 5x + 6 \).

### Step 2: Rearrange the equation
Rearranging gives us:
\[ x^2 - 5x + (6 - y) = 0 \]
This is a quadratic equation in \( x \).

### Step 3: Identify coefficients
In the quadratic equation \( ax^2 + bx + c = 0 \), we identify:
- \( a = 1 \)
- \( b = -5 \)
- \( c = 6 - y \)

### Step 4: Use the discriminant condition
For the quadratic equation to have real solutions, the discriminant must be non-negative:
\[ D = b^2 - 4ac \geq 0 \]
Substituting the values of \( a \), \( b \), and \( c \):
\[ (-5)^2 - 4 \cdot 1 \cdot (6 - y) \geq 0 \]

### Step 5: Simplify the discriminant
Calculating the discriminant:
\[ 25 - 4(6 - y) \geq 0 \]
Expanding:
\[ 25 - 24 + 4y \geq 0 \]
This simplifies to:
\[ 1 + 4y \geq 0 \]

### Step 6: Solve for \( y \)
Rearranging gives:
\[ 4y \geq -1 \]
Dividing by 4:
\[ y \geq -\frac{1}{4} \]

### Step 7: State the range
Thus, the range of \( f(x) \) is:
\[ y \in \left[-\frac{1}{4}, \infty\right) \]