3. Prove that the function f: R – {3} → R - {1} defined by f(x) = , is one-to-one correspondence.

Let f1(x)=3x and f2(x)=|x|. Graph both functions on the interval 2x2. Show that these functions are linearly dependent in the vector space C[0,1], but linearly independent in C[1,1].

Determine whether the function f : R → R given by f (x) = 4x + 6 is a bijective function.

(d) Suppose that ƒ: P → Q and g: Q→ R are functions. (i) Show that the composite function gof: P→ R is bijective if both f and g are bijective. (ii) If gof is bijective, show that (gof)-¹ = f-¹og-¹.

x + 1 (b) Let f(x) = and g(x) = x2 + 1. x + 2 ii. Determine if f and g are one to one or onto. iii. Determine if f og or go f are one to one or onto.

Let the function f: R R be defincd by f(r) = 3z - 5. Prove that the function is bijective.

2. Let f(x)=x2x3 - 3. a) Find the domain of the function: ○ [-3,00) Ⓒ (-∞0, 3] (-∞0, 3) U (3, 0) (∞, -3) U (-3,00) b) Find the range of the function: (-∞0, 2) U (2,00) ○ [2,00) Ⓒ (-∞, -2] O (-∞, -2) U (-2, 0) c) Is f a one-to-one function? Yes, f is one-to-one. No, f is not one-to-one. d) Use your answer from part c to determine if the inverse of the function f is a function. Yes, the inverse of the function f is a function. No, the inverse of the function f is not a function.

1 ab

Let f : R → R be defined as f(x) = x3 + 8(i) Prove that f is a bijection (one-to-one correspondence)(ii) Find the inverse of f.

5x+1 14. Prove that the function f: R \ {2} → R \ {5} defined by f(x) is bijective. x-2