Solved Math Problems - Functions, Equations, Logarithms

Q 1.
Answer: Option C
Figure 1: Domain of function Figure 2: Range of function
Domain: It is the interval over which the function is defined.
As seen in figure-1, the function is defined over the interval [-1,3], which is
therefore the domain.
Range: It is the range of values which the function assumes over the do-
main.
As seen in figure-2, the function takes values in-between -3 and 1, therefore,
the range is [-3,1].
Q 2.
Answer: Option C
√24 − 5x = x
[Square both sides]
24 − 5x = x2
[Subtract x2 from both sides]
−x2 − 5x + 24 = 0
[Multiply throughout by -1]
1

x2 + 5x − 24 = 0
=⇒ x2 + 8x − 3x − 24 = 0
x(x + 8) − 3(x + 8) = 0
(x + 8)(x − 3) = 0
∴ x = −8 or 3
Q 3.
Answer: Option C
Minima of the function occurs at the stationary point x = −2 and Maxima
occurs at the stationary point x = 4, as seen in the figure above.
The funciton decreases upto the minima and then increases from minima
to maxima. It again decreases once maxima is achieved. Therefore, the
function increases only in the interval (−2, 4).
Q 4.
Answer: Option D
A function f (x) is symmetric:
About y- axis if: f (x) = f (−x)
About x- axis if: f (x) = −f (x)
About origin if: f (x) = −f (−x)
For f (x) = |3 + x|,
f (−x) = |3 − x| 6 = f (x)
−f (x) = −|3 + x| 6 = f (x)
2

−f (−x) = −|3 − x| 6 = f (x)
Therefore, the function is not symmetric about any of the above.
Q 5.
Answer: Option B
|7 − 4x| ≥ 9
=⇒ 7 − 4x ≥ 9 or 7 − 4x ≤ −9
1.) 7 − 4x ≥ 9
[Subtract 7 from both sides]
−4x ≥ 2
[Divide throughout by −4]
=⇒ x ≤ − 1
2
2.) 7 − 4x ≤ −9
[Subtract 7 from both sides]
−4x ≤ −16
[Divide throughout by −4]
x ≥ 4
Therefore, the solution interval is
(−∞, −0.5] ∪ [4, ∞)
Q 6.
Answer: Option A
4x − 7y = 28
Arrange the equation of line in slope-intercept form
y = 4
7 x − 4
3