Advertisements
Advertisements
Question
Let A = {−2, −1, 0, 1, 2} and f : A → Z be a function defined by f(x) = x2 − 2x − 3. Find:
(b) pre-images of 6, −3 and 5.
Solution
(b) Let x be the pre-image of 6.
Then,
f(6) = x2 − 2x − 3 = 6
⇒ x2 − 2x − 9 = 0
⇒ \[x = 1 \pm \sqrt{10}\]
Since
Let x be the pre-image of -3. Then,
f(− 3) ⇒ x2 − 2x − 3 = − 3
⇒ x2 − 2x = 0
⇒ x = 0, 2
Clearly
So, 0 and 2 are pre-images of −3.
Let x be the pre-image of 5. Then,
f(5) ⇒ x2 − 2x − 3 = 5
⇒ x2 − 2x − 8 = 0
⇒ (x − 4) (x + 2) = 0 ⇒ x = 4, − 2
Since
Hence,
pre-images of 6, − 3 and 5 are \[\phi, \left\{ 0, 2, \right\}, - 2\] respectively.
APPEARS IN
RELATED QUESTIONS
Let f be the subset of Z × Z defined by f = {(ab, a + b): a, b ∈ Z}. Is f a function from Z to Z: justify your answer.
What is the fundamental difference between a relation and a function? Is every relation a function?
f, g, h are three function defined from R to R as follow:
(iii) h(x) = x2 + 1
Find the range of function.
If \[f\left( x \right) = \frac{1}{1 - x}\] , show that f[f[f(x)]] = x.
If \[f\left( x \right) = \frac{x + 1}{x - 1}\] , show that f[f[(x)]] = x.
If f(x) = (a − xn)1/n, a > 0 and n ∈ N, then prove that f(f(x)) = x for all x.
Let f : [0, ∞) → R and g : R → R be defined by \[f\left( x \right) = \sqrt{x}\] and g(x) = x. Find f + g, f − g, fg and \[\frac{f}{g}\] .
If f, g, h are real functions given by f(x) = x2, g(x) = tan x and h(x) = loge x, then write the value of (hogof)\[\left( \sqrt{\frac{\pi}{4}} \right)\] .
Write the domain and range of \[f\left( x \right) = \sqrt{x - \left[ x \right]}\] .
If 2f (x) − \[3f\left( \frac{1}{x} \right) = x^2\] (x ≠ 0), then f(2) is equal to
The function f : R → R is defined by f(x) = cos2 x + sin4 x. Then, f(R) =
If f : [−2, 2] → R is defined by \[f\left( x \right) = \begin{cases}- 1, & \text{ for } - 2 \leq x \leq 0 \\ x - 1, & \text{ for } 0 \leq x \leq 2\end{cases}\] , then
{x ∈ [−2, 2] : x ≤ 0 and f (|x|) = x} =
If \[e^{f\left( x \right)} = \frac{10 + x}{10 - x}\] , x ∈ (−10, 10) and \[f\left( x \right) = kf\left( \frac{200 x}{100 + x^2} \right)\] , then k =
If \[f\left( x \right) = 64 x^3 + \frac{1}{x^3}\] and α, β are the roots of \[4x + \frac{1}{x} = 3\] . Then,
Check if the relation given by the equation represents y as function of x:
x + y2 = 9
If f(x) = `("a" - x)/("b" - x)`, f(2) is undefined, and f(3) = 5, find a and b
Check the injectivity and surjectivity of the following function.
f : Z → Z given by f(x) = x2
Check the injectivity and surjectivity of the following function.
f : R → R given by f(x) = x2
Express the following logarithmic equation in exponential form
`log_(1/2) (8)` = – 3
Express the following logarithmic equation in exponential form
ln 1 = 0
Write the following expression as sum or difference of logarithm
`log (sqrt(x) root(3)(y))`
Prove that alogcb = blogca
If f(x) = 3x + 5, g(x) = 6x − 1, then find (fg) (3)
If f(x) = 3x + 5, g(x) = 6x − 1, then find `("f"/"g") (x)` and its domain
Select the correct answer from given alternatives.
If log10(log10(log10x)) = 0 then x =
Answer the following:
Let f : R – {2} → R be defined by f(x) = `(x^2 - 4)/(x - 2)` and g : R → R be defined by g(x) = x + 2. Examine whether f = g or not
Answer the following:
Let f : R → R be given by f(x) = x + 5 for all x ∈ R. Draw its graph
Answer the following:
Simplify, log (log x4) – log (log x)
Answer the following:
If `log ((x - y)/5) = 1/2 logx + 1/2 log y`, show that x2 + y2 = 27xy
Answer the following:
If `log_2"a"/4 = log_2"b"/6 = log_2"c"/(3"k")` and a3b2c = 1 find the value of k
Answer the following:
Find the domain of the following function.
f(x) = x!
Given the function f: x → x2 – 5x + 6, evaluate f(– 1)
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)
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
A function f is defined by f(x) = 2x – 3 find x such that f(x) = f(1 – x)
If f(x) = `x^3 - 1/x^3`, then `f(x) + f(1/x)` is equal to ______.
Domain of `sqrt(a^2 - x^2) (a > 0)` is ______.
If f : R – {2} `rightarrow` R i s a function defined by f(x) = `(x^2 - 4)/(x - 2)`, then its range is ______.
