Question

6. (a) Let \( f: \mathbb{R} \rightarrow \mathbb{R} \) be a function defined by \( f(x)=\frac{x^{2}-2}{x} \). (i) Find the Domain and Range of the function \( f \). (ii) Show that the function defined above is injective (one-to-one). (iii) Using the range obatined in (i), find the inverse \( f^{-1} \) of the function. (b) For each of the following determine the value(s) of the integer \( n>1 \) for which the given congruence is true (i) \( 28 \equiv 6(\bmod n) \). (ii) \( 68 \equiv 37(\bmod n) \).

          6. (a) Let \( f: \mathbb{R} \rightarrow \mathbb{R} \) be a function defined by \( f(x)=\frac{x^{2}-2}{x} \).
(i) Find the Domain and Range of the function \( f \).
(ii) Show that the function defined above is injective (one-to-one).
(iii) Using the range obatined in (i), find the inverse \( f^{-1} \) of the function.
(b) For each of the following determine the value(s) of the integer \( n>1 \) for which the given congruence is true
(i) \( 28 \equiv 6(\bmod n) \).
(ii) \( 68 \equiv 37(\bmod n) \).

6. (a) Let f: ℝ→ℝ be a function defined by f(x)=(x^2-2)/(x).
(i) Find the Domain and Range of the function f.
(ii) Show that the function defined above is injective (one-to-one).
(iii) Using the range obatined in (i), find the inverse f^-1 of the function.
(b) For each of the following determine the value(s) of the integer n>1 for which the given congruence is true
(i) 28 ≡ 6( n).
(ii) 68 ≡ 37( n).

Added by Selasi J.

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