If `f(x)=2x+7andg(x)=x^(2)+7,x""inR`, then what are values of x for which fog(x)=25?

Options

A. `-1,1`
B. `-2,2`
C. `-sqrt2,sqrt2`
D. None of these

Explanation

To solve the problem, we need to find the values of \( x \) for which \( f(g(x)) = 25 \), given the functions \( f(x) = 2x + 7 \) and \( g(x) = x^2 + 7 \).

### Step-by-Step Solution:

1. **Identify the Functions**:
- We have \( f(x) = 2x + 7 \)
- We have \( g(x) = x^2 + 7 \)

2. **Find \( f(g(x)) \)**:
- To find \( f(g(x)) \), we substitute \( g(x) \) into \( f(x) \):
\[ f(g(x)) = f(x^2 + 7) \]
- Now substitute \( g(x) \) into \( f(x) \):
\[ f(g(x)) = 2(x^2 + 7) + 7 \]

3. **Simplify \( f(g(x)) \)**:
- Expand the expression:
\[ f(g(x)) = 2x^2 + 14 + 7 = 2x^2 + 21 \]

4. **Set the Equation Equal to 25**:
- We need to solve the equation:
\[ 2x^2 + 21 = 25 \]

5. **Solve for \( x \)**:
- Subtract 21 from both sides:
\[ 2x^2 = 25 - 21 \]
\[ 2x^2 = 4 \]
- Divide both sides by 2:
\[ x^2 = 2 \]
- Take the square root of both sides:
\[ x = \pm \sqrt{2} \]

6. **Final Values of \( x \)**:
- The values of \( x \) for which \( f(g(x)) = 25 \) are:
\[ x = \sqrt{2} \quad \text{and} \quad x = -\sqrt{2} \]

### Summary:
The values of \( x \) for which \( f(g(x)) = 25 \) are \( x = \sqrt{2} \) and \( x = -\sqrt{2} \).