If `f(x)=x^(2)+1` and `f(g(2))=1`, then which of the following could be a possible expression for g(x)?C

Options

A. 5x-1
B. 2x-4
C. 6x+3
D. x-1

Answer

To solve the problem, we need to analyze the function \( f(x) = x^2 + 1 \) and the condition \( f(g(2)) = 1 \). Our goal is to find a possible expression for \( g(x) \) from the given options.

### Step-by-Step Solution:

1. **Understanding the function \( f(x) \)**:
\[ f(x) = x^2 + 1 \]
This function outputs a value that is always greater than or equal to 1, since \( x^2 \) is non-negative for all real \( x \).

2. **Setting up the equation**:
We know that \( f(g(2)) = 1 \). For \( f(x) \) to equal 1:
\[ x^2 + 1 = 1 \]
This simplifies to:
\[ x^2 = 0 \quad \Rightarrow \quad x = 0 \]
Therefore, we need \( g(2) = 0 \).

3. **Evaluating the options for \( g(x) \)**:
We will evaluate each option to see which one gives \( g(2) = 0 \).

- **Option A**: \( g(x) = 5x - 1 \)
\[ g(2) = 5(2) - 1 = 10 - 1 = 9 \]
(Not equal to 0)

- **Option B**: \( g(x) = 2x - 4 \)
\[ g(2) = 2(2) - 4 = 4 - 4 = 0 \]
(This is equal to 0)

- **Option C**: \( g(x) = 6x + 3 \)
\[ g(2) = 6(2) + 3 = 12 + 3 = 15 \]
(Not equal to 0)

- **Option D**: \( g(x) = x - 1 \)
\[ g(2) = 2 - 1 = 1 \]
(Not equal to 0)

4. **Conclusion**:
The only option that satisfies \( g(2) = 0 \) is **Option B**: \( g(x) = 2x - 4 \).

### Final Answer:
The possible expression for \( g(x) \) is:
\[ \boxed{2x - 4} \]