Advertisements
Advertisements
Question
If f(x) = 4x − x2, x ∈ R, then write the value of f(a + 1) −f(a − 1).
Solution
Given:
f(x) = 4x − x2, x ∈ R
Now,
f(a + 1) = 4(a + 1) -(a + 1)2
= 4a + 4 -(a2 + 1 + 2a)
= 4a + 4 -a2 -1 - 2a
= 2a -a2 + 3
f(a -1) = 4(a -1) - 1) +1)2
= 4a-4 - (a2 + 1 -2a)
= 4a - 4 - a2 -1 + 2a
= 6a - a2 -5
Thus,
f(a + 1) − f(a − 1) = ( 2a -a2 + 3) -(6a -a2 -5)
= 2a -a2 + 3 -6a + a2 + 5
= 8 -4a
= 4(2 -a)
APPEARS IN
RELATED QUESTIONS
Let A = {9, 10, 11, 12, 13} and let f: A → N be defined by f(n) = the highest prime factor of n. Find the range of f.
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.
If \[f\left( x \right) = x^3 - \frac{1}{x^3}\] , show that
If f(x) = loge (1 − x) and g(x) = [x], then determine function:
(i) f + g
If\[f\left( x \right) = 1 - \frac{1}{x}\] , then write the value of \[f\left( f\left( \frac{1}{x} \right) \right)\]
Write the domain and range of the function \[f\left( x \right) = \frac{x - 2}{2 - x}\] .
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)\[\left( \sqrt{\frac{\pi}{4}} \right)\] .
Find the set of values of x for which the functions f(x) = 3x2 − 1 and g(x) = 3 + x are equal.
If \[f\left( x \right) = \log \left( \frac{1 + x}{1 - x} \right)\] , then \[f\left( \frac{2x}{1 + x^2} \right)\] is equal to
If 2f (x) − \[3f\left( \frac{1}{x} \right) = x^2\] (x ≠ 0), then f(2) is equal to
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 \[3f\left( x \right) + 5f\left( \frac{1}{x} \right) = \frac{1}{x} - 3\] for all non-zero x, then f(x) =
The range of the function \[f\left( x \right) = \frac{x + 2}{\left| x + 2 \right|}\],x ≠ −2 is
The range of \[f\left( x \right) = \frac{1}{1 - 2\cos x}\] is
If f(m) = m2 − 3m + 1, find f(0)
Find x, if g(x) = 0 where g(x) = `(5x - 6)/7`
Find x, if g(x) = 0 where g(x) = x3 − 2x2 − 5x + 6
Check the injectivity and surjectivity of the following function.
f : N → N given by f(x) = x2
Check the injectivity and surjectivity of the following function.
f : N → N given by f(x) = x3
Solve for x.
log2 + log(x + 3) – log(3x – 5) = log3
Select the correct answer from given alternatives.
If f(x) =`1/(1 - x)`, then f{f[f(x)]} is
Answer the following:
A function f : R → R defined by f(x) = `(3x)/5 + 2`, x ∈ R. Show that f is one-one and onto. Hence find f–1
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) = 5–xPx–1
Answer the following:
Find the range of the following function.
f(x) = `x/(9 + x^2)`
Answer the following:
Find the range of the following function.
f(x) = [x] – x
A graph representing the function f(x) is given in it is clear that f(9) = 2
For what value of x is f(x) = 1?
An open box is to be made from a square piece of material, 24 cm on a side, by cutting equal square from the corner and turning up the side as shown. Express the volume V of the box as a function of 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
The range of the function f(x) = `(x^2 - 3x + 2)/(x^3 - 4x^2 + 5x - 2)` is ______
The domain of the function f(x) = log3+x (x2 - 1) is ______.
The range of the function y = `1/(2 - sin3x)` is ______.
The domain of the function f(x) = `1/sqrt(|x| - x)` is ______.
The function f: R `rightarrow` R defined by f(x) = sin x is ______.
If f : R – {2} `rightarrow` R i s a function defined by f(x) = `(x^2 - 4)/(x - 2)`, then its range is ______.
lf f : [0, ∞) `rightarrow` [0, ∞) and f(x) = `x/(1 + x)`, then f is ______.
The domain of f(x) = `sin^-1 [log_2(x/2)]` is ______.
