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The problem states the mapping is X 2x2 - x + 3, and asks to find the value of a. However, the variable a is not present in the given mapping. This indicates a likely typo in the question.

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b)
The problem states the mapping is X→2x2−x+3X \to 2x^2 - x + 3, and asks to find the value of aa. However, the variable aa is not present in the given mapping. This indicates a likely typo in the question. A common interpretation in such cases is that one of the coefficients in the quadratic expression was intended to be aa.

  1. Assume the mapping form.

    We will assume that the coefficient of x2x^2 was intended to be aa, making the mapping f(x)=ax2−x+3f(x) = ax^2 - x + 3.

  2. Use the first given condition to find aa.

    We are given that the image of 1 is 4, which means f(1)=4f(1) = 4.
    Substitute x=1x=1 into our assumed mapping:

    f(1)=a(1)2−(1)+3f(1) = a(1)^2 - (1) + 3 4=a−1+34 = a - 1 + 3 4=a+24 = a + 2

    Subtract 2 from both sides to solve for aa:

    a=4−2a = 4 - 2 a=2a = 2

  3. Verify the value of aa using the second condition.

    We are given that the image of 2 is 9, which means f(2)=9f(2) = 9.
    Substitute x=2x=2 and the calculated value a=2a=2 into our assumed mapping:

    f(2)=2(2)2−(2)+3f(2) = 2(2)^2 - (2) + 3 f(2)=2(4)−2+3f(2) = 2(4) - 2 + 3 f(2)=8−2+3f(2) = 8 - 2 + 3 f(2)=6+3f(2) = 6 + 3 f(2)=9f(2) = 9

    This matches the given condition, confirming our value of aa under the stated assumption.

    The value of aa is 2\boxed{2}.

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