If f(x) = 2x + 5 and g(x) = x² + 1 be two real functions, then describe each of the following functions:
(i) fog
(ii) gof
(iii) fof
(iv) f²

Also, show that fof ≠ f²

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Solution

f(x) and g(x) are polynomials.
$\Rightarrow$ f : R → R and g : R → R.
So, fog : R → R and gof : R → R.
$(i) (f o g) (x) = f (g (x)) = f (x^{2} + 1) = 2 (x^{2} + 1) + 5 = 2 x^{2} + 2 + 5 = 2 x^{2} + 7$

$(ii) (g o f) (x) = g (f (x)) = g (2 x + 5) = {(2 x + 5)}^{2} + 1 = 4 x^{2} + 20 x + 26$

$(iii) (f o f) (x) = f (f (x)) = f (2 x + 5) = 2 (2 x + 5) + 5 = 4 x + 10 + 5 = 4 x + 15$

$(iv) f^{2} (x) = f (x) \times f (x) = (2 x + 5) (2 x + 5) = {(2 x + 5)}^{2} = 4 x^{2} + 20 x + 25$

→→ →

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Let f, g be two real functions defined by $f (x) = \sqrt{x + 1}$ and $g (x) = \sqrt{9 - x^{2}}$ . Then, describe each of the following functions:

(i) $f + g$
(ii) $g - f$
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\frac{5}{8}

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