Which of the following functions is an even function?(a) \(f(x) = \fra

1.	Which of the following functions is an even function?(a) \(f(x) = \frac{a^x+1}{a^x -1}\) (b)\(f(x) = \frac{a^x- a^{-x}}{a^x +a^{-x}}\)(c) \(f(x) = x.\frac{a^x+1}{a^x -1}\)(d) f (x) = sin x
Answer» Answer: (c) \(f(x) = x.\frac{a^x+1}{a^x -1}\) (a) \(f(x) = \frac{a^x+1}{a^x -1}\) ∴ f (– x) = \( \frac{a^{-x}+1}{a^{-x} -1}\) = \(\frac{\frac{1}{a^x}+1}{\frac{1}{a^x} -1}\) = \(\frac{1+a^x}{1-a^x}\) = \(- \frac{a^x+1}{a^x-1}\) = - f(x) Hence f is odd. (b) Similarly \(f(x) = \frac{a^x-a^{-x}}{a^x +a^{-x}}\) is an odd function. (c) \(f(x) = x.\frac{a^x+1}{a^x -1}\) ∴ \(f(-x) = (-x)\frac{a^{-x}+1}{a^{-x} -1}\) = (-x) \(\frac{1/a^x +1}{1/a^x -1}\) = (-x) \(\frac{1+a^x}{1-a^x}\) = (-x) (\(- \frac{a^x+1}{a^x-1}\)) = x\(\big(\frac{a^x+1}{a^x-1}\big)\) = f(x) ⇒ f is even. (d) f (x) = sin x f (– x) = sin (– x) = – sin x = – f (x) ⇒ f is odd.

Which of the following functions is an even function?(a) \(f(x) = \frac{a^x+1}{a^x -1}\) (b)\(f(x) = \frac{a^x- a^{-x}}{a^x +a^{-x}}\)(c) \(f(x) = x.\frac{a^x+1}{a^x -1}\)(d) f (x) = sin x

Answer»

Answer: (c) \(f(x) = x.\frac{a^x+1}{a^x -1}\)

(a) \(f(x) = \frac{a^x+1}{a^x -1}\)

∴ f (– x) = \( \frac{a^{-x}+1}{a^{-x} -1}\) = \(\frac{\frac{1}{a^x}+1}{\frac{1}{a^x} -1}\) = \(\frac{1+a^x}{1-a^x}\)

= \(- \frac{a^x+1}{a^x-1}\) = - f(x)

Hence f is odd.

(b) Similarly \(f(x) = \frac{a^x-a^{-x}}{a^x +a^{-x}}\) is an odd function.

(c) \(f(x) = x.\frac{a^x+1}{a^x -1}\)

∴ \(f(-x) = (-x)\frac{a^{-x}+1}{a^{-x} -1}\)

= (-x) \(\frac{1/a^x +1}{1/a^x -1}\)

= (-x) \(\frac{1+a^x}{1-a^x}\) = (-x) (\(- \frac{a^x+1}{a^x-1}\))

= x\(\big(\frac{a^x+1}{a^x-1}\big)\) = f(x) ⇒ f is even.

(d) f (x) = sin x f (– x)

= sin (– x)

= – sin x = – f (x)

⇒ f is odd.