What values does the function `x rarr 2x^(2)-1` associate with the number 7 in the range?

Answer

To find the value that the function \( f(x) = 2x^2 - 1 \) associates with the number 7 in the range, we will follow these steps:

### Step 1: Define the function
The function is given as:
\[ f(x) = 2x^2 - 1 \]

### Step 2: Determine the domain of the function
The domain of the function is all real numbers since the function is defined for any real value of \( x \). Therefore, the domain is:
\[ (-\infty, \infty) \]

### Step 3: Find the minimum value of the function
To find the minimum value of \( f(x) \), we need to analyze the function. The term \( 2x^2 \) is always non-negative (since \( x^2 \geq 0 \)), and the minimum occurs when \( x = 0 \):
\[ f(0) = 2(0)^2 - 1 = -1 \]
Thus, the minimum value of the function is \(-1\).

### Step 4: Determine the range of the function
Since \( 2x^2 \) can take any non-negative value starting from 0, the function \( f(x) \) can take values starting from \(-1\) and going to positive infinity. Therefore, the range of the function is:
\[ [-1, \infty) \]

### Step 5: Find the value associated with the number 7
Now, we need to find \( f(7) \):
\[ f(7) = 2(7^2) - 1 = 2(49) - 1 = 98 - 1 = 97 \]
Thus, the value that the function associates with the number 7 is \( 97 \).

### Final Answer
The value that the function \( f(x) = 2x^2 - 1 \) associates with the number 7 is:
\[ \boxed{97} \]