Advertisements
Advertisements
Question
Find the domain and range of the following function.
f(x) = `sqrt(16 - x^2)`
Solution
f(x) = `sqrt(16 - x^2)`
For f to be defined,
16 – x2 ≥ 0
∴ x2 ≤ 16
∴ – 4 ≤ x < 4
∴ Domain of f = [– 4, 4]
Clearly, f(x) ≥ 0 and the value of f(x) would be maximum when the quantity subtracted from 16 is minimum i.e. x = 0
∴ Maximum value of f(x) = `sqrt(16)` = 4
∴ The range of f = [0, 4]
APPEARS IN
RELATED QUESTIONS
If \[f\left( x \right) = \begin{cases}x^2 , & \text{ when } x < 0 \\ x, & \text{ when } 0 \leq x < 1 \\ \frac{1}{x}, & \text{ when } x \geq 1\end{cases}\]
find: (a) f(1/2), (b) f(−2), (c) f(1), (d)
Let f and g be two real functions defined by \[f\left( x \right) = \sqrt{x + 1}\] and \[g\left( x \right) = \sqrt{9 - x^2}\] . Then, describe function:
(iii) f g
Let \[f\left( x \right) = \frac{\alpha x}{x + 1}, x \neq - 1\] . Then write the value of α satisfying f(f(x)) = x for all x ≠ −1.
Write the domain and range of \[f\left( x \right) = \sqrt{x - \left[ x \right]}\] .
Find the set of values of x for which the functions f(x) = 3x2 − 1 and g(x) = 3 + x are equal.
Let f and g be two real functions given by
f = {(0, 1), (2, 0), (3, −4), (4, 2), (5, 1)} and g = {(1, 0), (2, 2), (3, −1), (4, 4), (5, 3)}
Find the domain of fg.
If f(x) = cos (log x), then the value of f(x) f(y) −\[\frac{1}{2}\left\{ f\left( \frac{x}{y} \right) + f\left( xy \right) \right\}\] is
If \[f\left( x \right) = \log \left( \frac{1 + x}{1 - x} \right) \text{ and} g\left( x \right) = \frac{3x + x^3}{1 + 3 x^2}\] , then f(g(x)) is equal to
If A = {1, 2, 3} and B = {x, y}, then the number of functions that can be defined from A into B is
The function f : R → R is defined by f(x) = cos2 x + sin4 x. Then, f(R) =
The domain of the function \[f\left( x \right) = \sqrt{5 \left| x \right| - x^2 - 6}\] is
Check if the following relation is function:
Which of the following relations are functions? If it is a function determine its domain and range:
{(2, 1), (4, 2), (6, 3), (8, 4), (10, 5), (12, 6), (14, 7)}
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, 2), (2, −1), (3, 1), (4, 3)}
Find x, if f(x) = g(x) where f(x) = x4 + 2x2, g(x) = 11x2
Let f be a subset of Z × Z defined by f = {(ab, a + b) : a, b ∈ Z}. Is f a function from Z to Z? Justify?
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
54° = 1
Express the following logarithmic equation in exponential form
ln 1 = 0
Prove that logbm a = `1/"m" log_"b""a"`
If f(x) = 3x + 5, g(x) = 6x − 1, then find (fg) (3)
Answer the following:
If b2 = ac. prove that, log a + log c = 2 log b
Answer the following:
If `log"a"/(x + y - 2z) = log"b"/(y + z - 2x) = log"c"/(z + x - 2y)`, show that abc = 1
Let X = {3, 4, 6, 8}. Determine whether the relation R = {(x, f(x)) | x ∈ X, f(x) = x2 + 1} is a function from X to N?
A graph representing the function f(x) is given in it is clear that f(9) = 2
Describe the following Range
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 |
Check if this relation is a function
The function f and g are defined by f(x) = 6x + 8; g(x) = `(x - 2)/3`
Write an expression for gf(x) in its simplest form
Let A = {1, 2, 3, 4} and B = N. Let f : A → B be defined by f(x) = x3 then, find the range of f
If a function f(x) is given as f(x) = x2 – 6x + 4 for all x ∈ R, then f(–3) = ______.
Find the range of the following functions given by `sqrt(16 - x^2)`
Let A and B be any two sets such that n(B) = p, n(A) = q then the total number of functions f : A → B is equal to ______.
If f(x) = `(x - 1)/(x + 1)`, then show that `f(- 1/x) = (-1)/(f(x))`
Let f(x) = `sqrt(x)` and g(x) = x be two functions defined in the domain R+ ∪ {0}. Find (f – g)(x)
The domain and range of the function f given by f(x) = 2 – |x – 5| is ______.
Let f(θ) = sin θ (sin θ + sin 3θ) then ______.
