Advertisements
Advertisements
प्रश्न
Show that if f : A → B and g : B → C are one-one, then g ° f is also one-one
उत्तर
Let a, b ∈ A such that
(g ° f)(a) = (g ° f)(b)
∴ g[f(a)] = g[f(b)]
∴ f(a) = f(b) ...[∵ g is one-one]
∴ a = b ...[∵ f is one-one]
∴ (g ° f)(a) = (g ° f)(b) ⇒ a = b
∴ g ° f is one-one.
APPEARS IN
संबंधित प्रश्न
Which of the following relations are functions? Give reasons. If it is a function, determine its domain and range.
- {(2, 1), (5, 1), (8, 1), (11, 1), (14, 1), (17, 1)}
- {(2, 1), (4, 2), (6, 3), (8, 4), (10, 5), (12, 6), (14, 7)}
- {(1, 3), (1, 5), (2, 5)}
Let f : R+ → R, where R+ is the set of all positive real numbers, such that f(x) = loge x. Determine
(c) whether f(xy) = f(x) : f(y) holds
If \[f\left( x \right) = \frac{x + 1}{x - 1}\] , show that f[f[(x)]] = x.
If \[f\left( x \right) = \frac{2x}{1 + x^2}\] , show that f(tan θ) = sin 2θ.
If f is a real function satisfying \[f\left( x + \frac{1}{x} \right) = x^2 + \frac{1}{x^2}\]
for all x ∈ R − {0}, then write the expression for f(x).
Write the range of the function f(x) = cos [x], where \[\frac{- \pi}{2} < x < \frac{\pi}{2}\] .
Write the range of the function f(x) = ex−[x], x ∈ R.
If f(x) = 4x − x2, x ∈ R, then write the value of f(a + 1) −f(a − 1).
Let f(x) = x, \[g\left( x \right) = \frac{1}{x}\] and h(x) = f(x) g(x). Then, h(x) = 1
The range of the function f(x) = |x − 1| is
If ƒ(m) = m2 − 3m + 1, find f(x + 1)
If f(m) = m2 − 3m + 1, find f(− x)
Check if the following relation is a function.
Which sets of ordered pairs represent functions from A = {1, 2, 3, 4} to B = {−1, 0, 1, 2, 3}? Justify.
{(1, 3), (4, 1), (2, 2)}
Find x, if g(x) = 0 where g(x) = x3 − 2x2 − 5x + 6
Find the domain and range of the following function.
f(x) = `sqrt(16 - x^2)`
Check the injectivity and surjectivity of the following function.
f : R → R given by f(x) = x2
Express the following exponential equation in logarithmic form
e2 = 7.3890
Express the following logarithmic equation in exponential form
ln e = 1
Write the following expression as sum or difference of logarithm
`log (sqrt(x) root(3)(y))`
Write the following expression as sum or difference of logarithm
In `(("a"^3 ("a" - 2)^2)/sqrt("b"^2 + 5))`
Write the following expression as a single logarithm.
5 log x + 7 log y − log z
The equation logx2 16 + log2x 64 = 3 has,
Answer the following:
Identify the following relation is the function? If it is a function determine its domain and range.
{(2, 1), (4, 2), (6, 3), (8, 4), (10, 5), (12, 6), (14, 7)}
Answer the following:
Find whether the following function is one-one
f : R → R defined by f(x) = x2 + 5
Answer the following:
If f(x) = ax2 + bx + 2 and f(1) = 3, f(4) = 42, find a and b
Given the function f: x → x2 – 5x + 6, evaluate f(2a)
A graph representing the function f(x) is given in it is clear that f(9) = 2
Find the following values of the function
(a) f(0)
(b) f(7)
(c) f(2)
(d) f(10)
A function f is defined by f(x) = 2x – 3 find x such that f(x) = 0
The data in the adjacent table depicts the length of a person's forehand and their corresponding height. Based on this data, a student finds a relationship between the height (y) and the forehand length (x) as y = ax + b, where a, b are constant.
| Length ‘x’ of forehand (in cm) |
Height 'y' (in inches) |
| 35 | 56 |
| 45 | 65 |
| 50 | 69.5 |
| 55 | 74 |
Find the height of a person whose forehand length is 40 cm
The range of the function f(x) = `(x - 3)/(5 - x)`, x ≠ 5 is ______.
If a function f(x) is given as f(x) = x2 – 6x + 4 for all x ∈ R, then f(–3) = ______.
Let f : R → R be defined by
f(x) = `{(3x; x > 2),(2x^2; 1 ≤ x ≤ 2), (4x; x < 1):}`
Then f(-2) + f(1) + f(3) is ______
Find the domain for which the functions f(x) = 2x2 – 1 and g(x) = 1 – 3x are equal.
Find the domain of the following functions given by f(x) = x|x|
If f(x) = `(x - 1)/(x + 1)`, then show that `f(- 1/x) = (-1)/(f(x))`
Let f(θ) = sin θ (sin θ + sin 3θ) then ______.
The period of the function
f(x) = `(sin 8x cos x - sin 6x cos 3x)/(cos 2x cos x - sin 3x sin 4x)` is ______.
The domain of f(x) = `sin^-1 [log_2(x/2)]` is ______.
