All Functions (30-1)

Question 1

Determine the domain and range of f(x) = (x^2 - 4)/(x - 2).

Answer 1

Domain: x ≠ 2; Range: all real numbers except y = 4.

Question 2

Given f(x) = x^3 - 3x and g(x) = 2^x, find (f∘g)(x).

Answer 2

(f∘g)(x) = f(g(x)) = (2^x)^3 - 3(2^x) = 8^x - 3·2^x.

Question 3

Determine if f(x) = log(x - 2) and g(x) = x^2 are inverses.

Answer 3

Not inverses (both compositions ≠ x).

Question 4

Find the inverse of f(x) = 3^x + 2.

Answer 4

f⁻¹(x) = log3(x - 2).

Question 5

Sketch f(x) = 1/(x^2 - 1).

Answer 5

Vertical asymptotes x = ±1; horizontal asymptote y = 0; positive in (-∞,-1) and (1,∞), negative in (-1,1).

Question 6

Sketch f(x) = |2x - 3| - 1.

Answer 6

Vertex at (3/2, -1), slopes ±2, V-shape, domain all real numbers, range y ≥ -1.

Question 7

Determine the horizontal asymptote of f(x) = (3x^2 + 1)/(2x^2 - 5x + 2).

Answer 7

HA y = 3/2.

Question 8

Determine vertical asymptotes for f(x) = (x^2 - 4)/(x^2 - x - 6).

Answer 8

VA at x = 3, x = -2.

Question 9

Determine if f(x) = e^(x-1) - 2 and g(x) = ln(x+2) are inverses.

Answer 9

Yes, f⁻¹(x) = ln(x + 2) = g(x).

Question 10

Determine the vertex and axis of symmetry for f(x) = -x^2 + 4x - 1.

Answer 10

Vertex (2,3); axis x = 2.

Question 11

Solve |x^2 - 4| = 3.

Answer 11

Solutions: x = ±1, ±√7.

Question 12

Determine the end behaviour of f(x) = x^4 - 3x^3 + x - 5.

Answer 12

Both ends up.

Question 13

Sketch f(x) = log(x-1) + 2.

Answer 13

Vertical asymptote x = 1; domain x > 1, range all real numbers.

Question 14

Given f(x) = 2^x - 3 and g(x) = log2(x + 3), verify f and g are inverses.

Answer 14

Inverses confirmed.

Question 15

Match the function to the graph: cubic with roots at x = -2, 0, 3; left end down, right end up.

Answer 15

f(x) = a(x+2)x(x-3), a > 0.

Question 16

Given f(x) = (x-1)/(x+2), describe transformations from y = 1/x.

Answer 16

Horizontal shift left 2, vertical asymptote x = -2, horizontal asymptote y = 1.

Question 17

Determine y-intercept of f(x) = (x^2 - 4)/(x - 2).

Answer 17

y-intercept (0,2).

Question 18

Solve e^(2x) = 7.

Answer 18

x = ln7 / 2.

Question 19

Solve log3(x - 1) + log3(x + 2) = 2.

Answer 19

Solutions: (-1 + √45)/2, (-1 - √45)/2; only (-1 + √45)/2 valid.