rajlal wrote:

The functions \(f(x)\) and \(g(x)\) are defined by \(f(x) = x^2 – 1\) and \(g(x) = 1 – 2x\). Given that \(f(g(k)) = 3\), which of the following could be the value of k?

Options

A. \(\frac{1}{2}\)

B. \(\frac{√3}{2}\)

C. \(1\)

D. \(\frac{3}{2}\)

E.\(-1\)

Why does the method of substitution give me the wrong answer?

Given \(f(x) = x^2 - 1\) and \(g(x)= 1 - 2x\)

Now

\(g(k) = 1 - 2k\)

\(f(g(k)) = (1 - 2k)^2 - 1\)

But \(f(g(k)) = 3\)

so,

\(3 = (1 - 2k)^2 - 1\)

or \(4k^2 - 4k - 3 = 0\)

or \(4k^2 -6k + 2k - 3 = 0\)

or \(k^2 - \frac{3k}{2} +\frac{k}{2} - \frac{3}{4} = 0\)

or \((k - \frac{3}{2})(k +\frac{1}{2}) = 0\)

we get \(k = -\frac{1}{2} or \frac{3}{2}\)

Only Option D is correct

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