Q82 / 180 · COMEDK 2025 (10 May Shift 2) · Mathematics
If a function $f:[2, \infty) \rightarrow R$ defined by $f(x)=x^2-4 x+5$, then the range of $f$ is
- A$R$
- B$[5, \infty)$
- C$[2, \infty)$
- Correct answerD$[1, \infty)$
Solution
The function is given by $f(x) = x^2 - 4x + 5$.
Completing the square, we have $f(x) = (x^2 - 4x + 4) + 1 = (x - 2)^2 + 1$.
The domain of the function is $[2, \infty)$.
For $x \in [2, \infty)$, the term $(x - 2)$ takes values in $[0, \infty)$.
Consequently, $(x - 2)^2$ takes values in $[0, \infty)$.
Adding 1 to both sides, $(x - 2)^2 + 1$ takes values in $[1, \infty)$.
Thus, the range of the function $f$ is $[1, \infty)$.
Answer: $[1, \infty)$
Completing the square, we have $f(x) = (x^2 - 4x + 4) + 1 = (x - 2)^2 + 1$.
The domain of the function is $[2, \infty)$.
For $x \in [2, \infty)$, the term $(x - 2)$ takes values in $[0, \infty)$.
Consequently, $(x - 2)^2$ takes values in $[0, \infty)$.
Adding 1 to both sides, $(x - 2)^2 + 1$ takes values in $[1, \infty)$.
Thus, the range of the function $f$ is $[1, \infty)$.
Answer: $[1, \infty)$
Hence, the correct answer is option: 'D'.