Key: (b>0, b\neq 1) If (b>1) → growth; if (0<b<1) → decay.
(f(x)=2x-5) (y=2x-5 \Rightarrow x=2y-5 \Rightarrow 2y=x+5 \Rightarrow y=\fracx+52) So (f^-1(x)=\fracx+52)
(t_n = ar^n-1) Sum of (n) terms: (S_n = \fraca(r^n-1)r-1, r\neq 1)
I cannot produce an entire (e.g., Nelson Functions 11 , McGraw-Hill Ryerson Functions 11 ) page-by-page, as that would violate copyright.
Period of sine/cosine: (360^\circ) ((2\pi) rad) Period of tangent: (180^\circ) ((\pi) rad)
Check: (f^-1(f(x)) = \frac2x-5+52 = x). General form: (f(x) = a\cdot b^k(x-d) + c)
(f(x)=x^2+1), (g(x)=2x-3) Find ((f\circ g)(x) = f(g(x)) = (2x-3)^2 + 1 = 4x^2 -12x + 10) 3. Transformations of Functions Given (y = a,f(k(x-d)) + c):
(0^\circ, 30^\circ, 45^\circ, 60^\circ, 90^\circ) and their radian equivalents.
(y = 3\cos(2x - \pi) + 1) Rewrite: (y = 3\cos(2(x - \pi/2)) + 1) Amplitude 3, Period (360/2=180^\circ) ((\pi) rad), Phase shift (\pi/2) right, Vertical shift 1 up. 8. Sequences & Series Arithmetic sequence: (t_n = a + (n-1)d) Sum of (n) terms: (S_n = \fracn2(2a + (n-1)d))
Start with (f(x)=x^2). Apply: vertical compression by (1/2), shift right 3, shift up 4. [ y = \frac12 (x-3)^2 + 4 ] 4. Inverse Functions Switch (x) and (y) in (y=f(x)), then solve for (y). Inverse exists if (f) is one‑to‑one (passes horizontal line test).
(y = a\sin(k(x-d)) + c) Amplitude = (|a|), Period = (360^\circ/|k|) (or (2\pi/|k|) rad), Phase shift = (d), Vertical shift = (c)
Below is a summary + original problems. Grade 11 Functions – Study Paper Topics: Characteristics of functions, domain/range, transformations, inverse functions, exponential functions, trigonometric functions, sequences & series. 1. Function Basics Definition: A function (f) pairs each element (x) in the domain with exactly one element (y) in the range.
Functions Grade 11 Textbook Page
Key: (b>0, b\neq 1) If (b>1) → growth; if (0<b<1) → decay.
(f(x)=2x-5) (y=2x-5 \Rightarrow x=2y-5 \Rightarrow 2y=x+5 \Rightarrow y=\fracx+52) So (f^-1(x)=\fracx+52)
(t_n = ar^n-1) Sum of (n) terms: (S_n = \fraca(r^n-1)r-1, r\neq 1)
I cannot produce an entire (e.g., Nelson Functions 11 , McGraw-Hill Ryerson Functions 11 ) page-by-page, as that would violate copyright. functions grade 11 textbook
Period of sine/cosine: (360^\circ) ((2\pi) rad) Period of tangent: (180^\circ) ((\pi) rad)
Check: (f^-1(f(x)) = \frac2x-5+52 = x). General form: (f(x) = a\cdot b^k(x-d) + c)
(f(x)=x^2+1), (g(x)=2x-3) Find ((f\circ g)(x) = f(g(x)) = (2x-3)^2 + 1 = 4x^2 -12x + 10) 3. Transformations of Functions Given (y = a,f(k(x-d)) + c): Key: (b>0, b\neq 1) If (b>1) → growth;
(0^\circ, 30^\circ, 45^\circ, 60^\circ, 90^\circ) and their radian equivalents.
(y = 3\cos(2x - \pi) + 1) Rewrite: (y = 3\cos(2(x - \pi/2)) + 1) Amplitude 3, Period (360/2=180^\circ) ((\pi) rad), Phase shift (\pi/2) right, Vertical shift 1 up. 8. Sequences & Series Arithmetic sequence: (t_n = a + (n-1)d) Sum of (n) terms: (S_n = \fracn2(2a + (n-1)d))
Start with (f(x)=x^2). Apply: vertical compression by (1/2), shift right 3, shift up 4. [ y = \frac12 (x-3)^2 + 4 ] 4. Inverse Functions Switch (x) and (y) in (y=f(x)), then solve for (y). Inverse exists if (f) is one‑to‑one (passes horizontal line test). General form: (f(x) = a\cdot b^k(x-d) + c)
(y = a\sin(k(x-d)) + c) Amplitude = (|a|), Period = (360^\circ/|k|) (or (2\pi/|k|) rad), Phase shift = (d), Vertical shift = (c)
Below is a summary + original problems. Grade 11 Functions – Study Paper Topics: Characteristics of functions, domain/range, transformations, inverse functions, exponential functions, trigonometric functions, sequences & series. 1. Function Basics Definition: A function (f) pairs each element (x) in the domain with exactly one element (y) in the range.
