The function f(x)=(x²+2 x-15)/(x²-4 x+9), x ∈ ℝ is

Solution

Maths sir · 4 steps

Two inputs give the same zero, so it is not one-one, and the range is a closed interval, so it is not onto.

1. Test one-one. The numerator factors as (x-3)(x+5), so f(3) = 0 and f(-5) = 0. Two different inputs give the same output, so f is not one-one.
2. Set up the range. The denominator x²-4x+9 has discriminant 16-36<0, so it is never zero and f is defined for every real x. Put f(x)=y and cross-multiply to get a quadratic in x.

   (y-1)x²-(4y+2)x+(9y+15)=0

3. Real x needed. For a real x the discriminant must satisfy (4y+2)²-4(y-1)(9y+15)≥ 0, which simplifies to 5y²+2y-16≤ 0, that is (y+2)(5y-8)≤ 0. So -2≤ y≤ 8/5. The case y=1 gives x=4, which lies in this interval anyway.
4. So. The range is [-2, 8/5], not all of ℝ, so f is not onto either. The answer is (c), neither one-one nor onto.

The idea to keep. For a rational function quadratic over quadratic on ℝ, set f(x)=y and demand a real x. The discriminant gives the range. Equal outputs at two inputs, such as two roots of the numerator, rule out one-one.

Answer from the NTA final answer key. Solution by PrepWiser.