What is the ranger of the function `f(x)=x^2-14+=34` ?

Options

A. `xle7`
B. `xge0`
C. `yle-6`
D. `yge-6`

Explanation

To find the range of the function \( f(x) = x^2 - 14x + 43 \), we will follow these steps:

### Step 1: Identify the function
We start with the function:
\[ f(x) = x^2 - 14x + 43 \]

### Step 2: Differentiate the function
To find the minimum value of the function, we need to differentiate it with respect to \( x \):
\[ f'(x) = \frac{d}{dx}(x^2 - 14x + 43) = 2x - 14 \]

### Step 3: Set the derivative to zero
Next, we set the derivative equal to zero to find the critical points:
\[ 2x - 14 = 0 \]

### Step 4: Solve for \( x \)
Solving for \( x \):
\[ 2x = 14 \implies x = \frac{14}{2} = 7 \]

### Step 5: Find the minimum value of \( f(x) \)
Now, we substitute \( x = 7 \) back into the original function to find the minimum value:
\[ f(7) = 7^2 - 14 \times 7 + 43 \]
Calculating this step-by-step:
\[ = 49 - 98 + 43 \]
\[ = -49 + 43 = -6 \]

### Step 6: Determine the range
Since \( f(x) \) is a quadratic function that opens upwards (the coefficient of \( x^2 \) is positive), the minimum value we found is the lowest point of the function. Thus, the range of \( f(x) \) is:
\[ f(x) \geq -6 \]
In interval notation, the range is:
\[ [-6, \infty) \]

### Final Answer
The range of the function \( f(x) = x^2 - 14x + 43 \) is:
\[ [-6, \infty) \]