How do we multiply f(x) with its inverse

Mathematics

ScanSolve AI Answer

Answer

x2−49x^2 - \frac{4}{9}

Step-by-step solution

3 steps

To multiply a function f(x)f(x) with its inverse f−1(x)f^{-1}(x), you first need to find both expressions and then perform the algebraic multiplication.

Let's use the example from our previous conversation:
f(x)=x−23f(x) = x - \frac{2}{3}
And its inverse:
f−1(x)=x+23f^{-1}(x) = x + \frac{2}{3}

  1. Write out the multiplication of f(x)f(x) and f−1(x)f^{-1}(x).

    f(x)⋅f−1(x)=(x−23)⋅(x+23)f(x) \cdot f^{-1}(x) = \left(x - \frac{2}{3}\right) \cdot \left(x + \frac{2}{3}\right)

  2. Use the difference of squares formula, (a−b)(a+b)=a2−b2(a-b)(a+b) = a^2 - b^2.

    Here, a=xa = x and b=23b = \frac{2}{3}.

    f(x)⋅f−1(x)=x2−(23)2f(x) \cdot f^{-1}(x) = x^2 - \left(\frac{2}{3}\right)^2

  3. Simplify the expression.

    f(x)⋅f−1(x)=x2−2232f(x) \cdot f^{-1}(x) = x^2 - \frac{2^2}{3^2} f(x)⋅f−1(x)=x2−49f(x) \cdot f^{-1}(x) = x^2 - \frac{4}{9}

    The product of f(x)f(x) and its inverse f−1(x)f^{-1}(x) is:

    x2−49x^2 - \frac{4}{9}
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