Question

1. The functions g, f: R → R are defined by f(x) = 5x − 1 and g(x) = 3x 2 + 4. Determine f ◦ g and g ◦ f.2. The functions g, f: R → R are defined by f(x) = 1 − 3x and g(x) = x 2 + 1. Evaluate f(g(f(0))).3. Describe g ◦ f. (a) f: Z → N, f(n) = n 2 + 1; g: N → Q, g(n) = 1 n . (b) f: R → (0, 1), f(x) = 1/(x 2 + 1); g: (0, 1) → (0, 1), g(x) = 1 − x. (c) f: Q − {2} → Q∗ , f(x) = 1/(x − 2); g: Q∗ → Q∗ , g(x) = 1/x. (d) f: R → [ 1, [infinity]), f(x) = x 2 + 1; g: [ 1, [infinity]) → [ 0, [infinity]) g(x) = √ x − 1. (e) f: Q − {10/3} → Q − {3}, f(x) = 3x − 7; g: Q − {3} → Q − {2}, g(x) = 2x/(x − 3).4. Give an example of sets A, B, and C, and of functions f: A → B and g: B → C, such that g ◦ f and f are both one-to-one, but g is not one-to-one.

Answer 1

We have f(x) = 5x - 1 and g(x) = 3x^2 + 4. To determine f ◦ g, we substitute g(x) into f(x) and get:

f(g(x)) = 5(3x^2 + 4) - 1

f(g(x)) = 15x^2 + 19

To determine g ◦ f, we substitute f(x) into g(x) and get:

g(f(x)) = (5x - 1)^2 + 1

g(f(x)) = 25x^2 - 10x + 2

We have f(x) = 1 - 3x and g(x) = x^2 + 1. We evaluate f(g(f(0))) as follows:

f(g(f(0))) = f(g(1)) (since f(0) = 1)

f(g(f(0))) = f(2) (since g(1) = 2)

f(g(f(0))) = 1 - 3(2) (since f(2) = -5)

f(g(f(0))) = -14

Therefore, f(g(f(0))) = -14.

(a) We have f(n) = n^2 + 1 and g(n) = 1/n. Then,

(g ◦ f)(n) = g(f(n)) = g(n^2 + 1) = 1/(n^2 + 2)

(b) We have f(x) = 1/(x^2 + 1) and g(x) = 1 - x. Then,

(g ◦ f)(x) = g(f(x)) = g(1/(x^2 + 1)) = 1 - 1/(x^2 + 1)

(c) We have f(x) = 1/(x - 2) and g(x) = 1/x. Then,

(g ◦ f)(x) = g(f(x)) = g(1/(x - 2)) = (x - 2)/x

(d) We have f(x) = x^2 + 1 and g(x) = sqrt(x - 1). Then,

(g ◦ f)(x) = g(f(x)) = sqrt(x^2) = x

(e) We have f(x) = 3x - 7 and g(x) = 2x/(x - 3). Then,

(g ◦ f)(x) = g(f(x)) = 2(3x - 7)/(3x - 10)

Let A = {1, 2}, B = {3, 4}, C = {5, 6}, f: A → B be defined by f(1) = 3, f(2) = 4, and g: B → C be defined by g(3) = g(4) = 5. Then g ◦ f is the function from A to C that maps 1 to 5 and 2 to 5. Both g ◦ f and f are one-to-one, but g is not one-to-one since it maps two elements in B to the same element in C.