Zadanie 3
Let f(x) = x2 + 2x and g(x) = ex − 6, both with real domain. Find all real x satisfying (f∘g)(x) = 24.
Composition and inverse functions
Combine functions in a specified order and determine inverses where the original mapping is one-to-one.
Tę umiejętność sprawdza 37 zadań wzorowanych na arkuszach IB z lat 2022–2026.
Zadania sprawdzające tę umiejętność: 37 Otwórz w wyszukiwarce z filtrami
Zadanie 6a
Find f⁻¹(x).
Zadanie 6b
A function g satisfies (f∘g)(2x) = x wherever defined. Find g(x).
Zadanie 12c
(i) Find f⁻¹(x) and its domain. (ii) Find the intersection of y=f(x) and y=f⁻¹(x) with x>0.
Zadanie 1a
Show that (i) f∘2(x) = c1+mxm²; (ii) f∘3(x) = c1+m+m²xm³.
Zadanie 1b
(i) State f∘4(x). (ii) Conjecture a corresponding expression for f∘n(x) for positive integers n. (iii) Use your expression, or another method, to find f∘n(x) when m = 1.
Zadanie 1c
For m ≠ 1, prove by mathematical induction that for every positive integer n.
Zadanie 1e
Now let m = −1. (i) Show that f∘n(x) = c/x when n is odd. (ii) Find f∘n(x) when n is even.
Zadanie 2d
The logarithmic profile used for comparison is v(x) = 2ln x, x > 0. For w(x) = ex/2, , you are given and . (i) Show that…
Zadanie 2b
State the domain of f−1.
- Materiał graficzny
Zadanie 2c
Solve f−1(4x − 3) = −2.
- Materiał graficzny
Zadanie 6b
Find f−1(3).
Zadanie 10d
(i) Write an expression for (f∘g)(x). (ii) Solve (f∘g)(x) ≤ 24.
Zadanie 10e
Find the domain of g∘f.
Zadanie 10a
Find f−1(9) as a fraction of integers.
Zadanie 10b
(i) Find g−1(x). (ii) Describe a sequence of transformations taking the graph of g−1 to the graph of f.
Zadanie 10c
Show that (f∘g)(x) = 729x3.
Zadanie 12d
Define g(x) = F(x) for 0 ≤ x < K. (i) Justify that g−1 exists. (ii) Find g−1(x) and its domain.
Zadanie 12e
Find the area of the shaded region bounded by portions of y = g(x), y = g−1(x) and the positive coordinate axes.
- Materiał graficzny
Zadanie 1a
On the same axes, sketch and label y=F₁(x) and y=F₂(x), showing all axis intercepts.
Zadanie 1b
Find the maximum values of (i) F₁; (ii) F₂; (iii) F₃.
Zadanie 1c
Find the least positive integer n for which the maximum of Fₙ is less than 0.45.
Zadanie 6b
Find the greatest k for which f is real-valued throughout x≤k and has an inverse on that domain. Justify your answer.
Zadanie 2a
Find (f∘g)(x).
Zadanie 5
Let f(x)=ax+b, where a,b are integers, and g(x)=x²−2x+5. Find both possible functions f for which (g∘f)(x) = 9x2 + 12x + 8 for all real x.
Zadanie 1a
Write an expression for (g∘f)(x).
Zadanie 1b
Given (g∘f)(1) = 25, find all possible k.
Zadanie 1a
Write an expression for (g∘f)(x).
Zadanie 1b
Given (g∘f)(1) = 25, find all possible k.
Zadanie 3a
Find (f∘g)(x).
Zadanie 10b
(i) Show that . (ii) State its domain and range.
Zadanie 11b
(i) Show that . (ii) State its domain.
Zadanie 11c
Given (h∘g)(a) = π/4, find a in exact simplified form.
Zadanie 8a
For k = π/2, find the largest a for which an inverse exists. A sketch or an algebraic argument may be used.
Zadanie 8b
Find the largest a when k = π.
Zadanie 8c
Find the largest a in terms of k when π < k < 3π/2.
Zadanie 11d
Find (i)(f∘g)(0); (ii)(f∘g)′(0).