Advertisements
Advertisements
Question
Let f : R → R be defined by f(x) = 2x + |x|. Then f(2x) + f(−x) − f(x) =
Options
(a) 2x
(b) 2|x|
(c) −2x
(d) −2|x|
Solution
(b) 2|x|
f(x) = 2x + |x|
Then, f(2x) + f(−x) − f(x)
\[= 2\left( 2x \right) + 2\left| x \right| + \left( - 2x \right) + \left| - x \right| - 2x + \left| x \right|\]
\[ = 4x - 2x - 2x + 2\left| x \right| + \left| - x \right| - \left| x \right|\]
\[ = 0 + 2\left| x \right| + \left| x \right| - \left| x \right|\]
\[ = 2\left| x \right|\]
APPEARS IN
RELATED QUESTIONS
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
Let f : R → R and g : C → C be two functions defined as f(x) = x2 and g(x) = x2. Are they equal functions?
If \[f\left( x \right) = \frac{1}{1 - x}\] , show that f[f[f(x)]] = x.
If for non-zero x, af(x) + bf \[\left( \frac{1}{x} \right) = \frac{1}{x} - 5\] , where a ≠ b, then find f(x).
If f, g and h are real functions defined by
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\[f\left( x \right) = 1 - \frac{1}{x}\] , then write the value of \[f\left( f\left( \frac{1}{x} \right) \right)\]
If f : Q → Q is defined as f(x) = x2, then f−1 (9) is equal to
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(x) = sin [π2] x + sin [−π]2 x, where [x] denotes the greatest integer less than or equal to x, then
Check if the following relation is function:
Check if the following relation is function:
If f(x) = 3x + a and f(1) = 7 find a and f(4).
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, 1), (2, 1), (3, 1), (4, 1)}
Check if the relation given by the equation represents y as function of x:
x + y2 = 9
If f(m) = m2 − 3m + 1, find `f(1/2)`
Find the domain and range of the following function.
f(x) = `sqrt((x - 2)(5 - x)`
Express the area A of circle as a function of its radius r
Express the area A of circle as a function of its diameter d
Express the following logarithmic equation in exponential form
ln e = 1
If f(x) = ax2 − bx + 6 and f(2) = 3 and f(4) = 30, find a and b
If `log(( x - y)/4) = logsqrt(x) + log sqrt(y)`, show that (x + y)2 = 20xy
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 log (5x – 9) – log (x + 3) = log 2 then x = ...............
Answer the following:
Simplify `log_10 28/45 - log_10 35/324 + log_10 325/432 - log_10 13/15`
Answer the following:
Show that `7log (15/16) + 6log(8/3) + 5log (2/5) + log(32/25)` = log 3
Answer the following:
Find value of `(3 + log_10 343)/(2 + 1/2 log_10 (49/4) + 1/2 log_10 (1/25)`
Answer the following:
If `log"a"/(x + y - 2z) = log"b"/(y + z - 2x) = log"c"/(z + x - 2y)`, show that abc = 1
Answer the following:
Find the domain of the following function.
f(x) = `sqrt(x - 3) + 1/(log(5 - x))`
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
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
Redefine the function which is given by f(x) = `|x - 1| + |1 + x|, -2 ≤ x ≤ 2`
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)` = – f(x)
The domain of the function f defined by f(x) = `sqrt(4 - x) + 1/sqrt(x^2 - 1)` is equal to ______.
The range of the function f(x) = `""^(7 - x)P_(x - 3)` is ______.
