Preparation Module: Functions (Round 2)

CSCA Preparation Module: Functions (Round 2)

Advanced Properties, Composite Functions, and Equations

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1. Summary of Concepts & Formulas

1.1 Concepts and Properties of Functions

• Domain ($D_{f}$):
  • Fraction $\frac{A}{B}$: Condition $B \ne 0$
  • Even Root $\sqrt[2n]{A}$: Condition $A \ge 0$
  • Logarithm $\log_{a}(A)$: Condition $A > 0$ and $a > 0$, $a \ne 1$
• Parity:
  • Even Function: $f(-x) = f(x)$. Symmetric to y-axis.
  • Odd Function: $f(-x) = -f(x)$. Symmetric to origin.
• Monotonicity:
  • Increasing: $x_{1} < x_{2} \Rightarrow f(x_{1}) < f(x_{2})$
  • Decreasing: $x_{1} < x_{2} \Rightarrow f(x_{1}) > f(x_{2})$

1.2 Elementary Functions

• Exponential ($y=a^{x}$): Domain $\mathbb{R}$, Range $(0, +\infty)$. Increasing if $a>1$, decreasing if $0
• Logarithmic ($y=\log_{a}x$): Domain $(0, +\infty)$, Range $\mathbb{R}$. Change of base: $\log_{a}b = \dfrac{\ln b}{\ln a}$.
• Trigonometry:
  • $\sin^2 x + \cos^2 x = 1$
  • $\sin(2x) = 2 \sin x \cos x$
  • $\cos(2x) = \cos^{2}x - \sin^{2}x$

2. Summary of Advanced Concepts

2.1 Composite and Inverse Functions

• Composite: $(f \circ g)(x) = f(g(x))$. Domain depends on inner function range meeting outer function domain.
• Inverse: $y=f(x)$ and $y=f^{-1}(x)$ are symmetric about $y=x$. Domain of $f^{-1}$ is Range of $f$.

2.2 Advanced Properties

• Even/Odd Operations:
  • Even $\cdot$ Odd = Odd
  • Odd $\cdot$ Odd = Even
  • Even $\cdot$ Even = Even
• Auxiliary Angle: $a \sin x + b \cos x = \sqrt{a^{2}+b^{2}} \sin(x+\phi)$, where $\tan \phi =\dfrac{b}{a}$.

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3. Practice Test (Round 2)

Part 1: Advanced Concepts (Composite, Inverse, Parity)

1. If $f(x)=2x+1$ and $g(x)=x^{2}$ find $(g \circ f)(x)$.
   1. $2x^{2}+1$
   2. $4x^{2}+1$
   3. $4x^{2}+4x+1$
   4. $2x^{2}+2x+1$
   Explanation $(g \circ f)(x) = g(f(x)) = g(2x+1)$
   $ = (2x+1)^{2} = 4x^{2}+4x+1$. Correct Answer: C
2. The function $f(x)=x^{3}+\sin x$ is:
   1. Even
   2. Odd
   3. Neither even nor odd
   4. Periodic
   Explanation $f(-x) = (-x)^{3} + \sin(-x) $
   $= -x^{3} - \sin x = -(x^{3} + \sin x) = -f(x)$. Thus, it is Odd. Correct Answer: B
3. Find the domain of $f(x)=\dfrac{1}{\ln(x-1)}$.
   1. $(1, +\infty)$
   2. $(1,2)\cup(2,+\infty)$
   3. $(2, +\infty)$
   4. $[2, +\infty)$
   Explanation 1. Log argument: $x-1 > 0 \Rightarrow x > 1$.
   2. Denominator: $\ln(x-1) \ne 0 \Rightarrow x-1 \ne 1 \Rightarrow x \ne 2$.
   Domain: $(1,2) \cup (2,+\infty)$. Correct Answer: B
4. If $f(x)=\dfrac{x}{1+x} ,(x>-1)$, find $f^{-1}(x)$.
   1. $\dfrac{x}{1-x}$
   2. $\dfrac{1+x}{x}$
   3. $\dfrac{x}{x-1}$
   4. $1-x$
   Explanation Let $y = \dfrac{x}{1+x} $
   $\Rightarrow y + xy = x \Rightarrow y = x - xy = x(1-y)$.
   $x = \frac{y}{1-y}$. Swap variables: $y = \frac{x}{1-x}$. Correct Answer: A
5. Given $f(x)$ is an even function on $\mathbb{R}$ and increasing on $[0,+\infty)$. Compare $f(-2)$ and $f(3)$.
   1. $f(-2)>f(3)$
   2. $f(-2) < f(3)$
   3. $f(-2)=f(3)$
   4. Cannot determine
   Explanation Since $f$ is even, $f(-2) = f(2)$.
   Since $f$ is increasing on $[0, +\infty)$ and $2 < 3$, then $f(2) < f(3)$.
   Therefore, $f(-2) < f(3)$. Correct Answer: B
6. If $f(x+1)=x^{2}-3x+2$, what is $f(x)$?
   1. $x^{2}-5x+6$
   2. $x^{2}-x+2$
   3. $x^{2}-3x$
   4. $x^{2}+x-2$
   Explanation Let $t = x+1 \Rightarrow x = t-1$.
   $f(t) = (t-1)^{2} - 3(t-1) + 2 $
   $= (t^{2} - 2t + 1) - 3t + 3 + 2 = t^{2} - 5t + 6$. Correct Answer: A
7. The range of $y=\dfrac{2^{x}-1}{2^{x}+1}$ is:
   1. $(-1,1)$
   2. $(0,1)$
   3. $(-1,0)$
   4. $\mathbb{R}$
   Explanation $y = \frac{2^{x}+1-2}{2^{x}+1} = 1 - \frac{2}{2^{x}+1}$.
   Since $2^{x} > 0$, denominator $2^{x}+1 \in (1,+\infty)$.
   Fraction $\frac{2}{2^{x}+1} \in (0,2)$.
   Range: $(1-2, 1-0) = (-1,1)$. Correct Answer: A
8. Which function is the inverse of itself?
   1. $y=x^{2}$
   2. $y=\frac{1}{x}$
   3. $y=e^{x}$
   4. $y=x+1$
   Explanation For $y=1/x$, swap $x$ and $y$: $x=1/y \Rightarrow y=1/x$. It is its own inverse. Correct Answer: B
9. If $f(x)=\ln(\sqrt{1+x^{2}}-x)$, then $f(x)$ is:
   1. Even
   2. Odd
   3. Neither
   4. Constant
   Explanation Note that $\sqrt{1+x^{2}}-x = \frac{1}{\sqrt{1+x^{2}}+x}$.
   $f(-x) = \ln(\sqrt{1+x^{2}}+x)$
   $ = \ln((\sqrt{1+x^{2}}-x)^{-1}) $
   $= -\ln(\sqrt{1+x^{2}}-x) = -f(x)$. Correct Answer: B
10. Find the domain of $y=\sqrt{1-\log_{2}x}$.
    1. $(0,2]$
    2. $[1,2]$
    3. $(0,1)$
    4. $[2,+\infty)$
    Explanation 1. Root: $1-\log_{2}x \ge 0 \Rightarrow \log_{2}x \le 1 \Rightarrow x \le 2$.
    2. Log: $x > 0$.
    Domain: $(0,2]$. Correct Answer: A
11. If $f(g(x))=x$ for all $x$, and $f(x)=3^{x}$, what is $g(9)$?
    1. 3
    2. 2
    3. 9
    4. $1/3$
    Explanation $f(g(x)) = x$ means $g(x)$ is the inverse of $f(x)$.
    $f(x) = 3^{x} \Rightarrow g(x) = \log_{3}x$.
    $g(9) = \log_{3}9 = 2$. Correct Answer: B
12. Let $A=\{x \mid x^{2}-2x-3<0\}$ and $B=\{x \mid y=\ln(x-1)\}$. Find $A \cap B$.
    1. $(1,3)$
    2. $(-1,3)$
    3. $(1,+\infty)$
    4. $(-1,1)$
    Explanation Set A: $(x-3)(x+1) < 0 \Rightarrow -1 < x < 3$.
    Set B (Domain): $x-1 > 0 \Rightarrow x > 1$.
    Intersection: $(1, 3)$. Correct Answer: A

Part 2: Power & Exponential Equations/Inequalities

13. Solve $4^{x}-2^{x+2}+3=0$.
    1. $x=\log_{2}3$
    2. $x=\log_{2}3, x=0$
    3. $x=3, x=1$
    4. $x=\log_{2}3$
    Explanation Rewrite: $(2^{x})^{2} - 4(2^{x}) + 3 = 0$. Let $u=2^{x}$.
    $(u-3)(u-1) = 0 \Rightarrow u=3$ or $u=1$.
    $2^{x}=3 \Rightarrow x=\log_{2}3$.
    $2^{x}=1 \Rightarrow x=0$. Correct Answer: B
14. If $a=0.3^{0.2}$, $b=0.3^{2}$, $c=2^{0.3}$, order $a, b, c$.
    1. $b < a < c $
    2. $a < b < c $
    3. $c < a < b $
    4. $b < c < a $
    Explanation Base $0.3 < 1$, so larger exponent means smaller value: $0.3^{2} < 0.3^{0.2}$ ($b < a$).
    $a = 0.3^{0.2} < 1$. $c = 2^{0.3} > 1$.
    Order: $b < a < c $. Correct Answer: A
15. Solve the inequality $2^{x^{2}-x} < 4$.
    1. $-1 < x < 2 $
    2. $ x<-1 $ or $ x > 2 $
    3. $ 1 < x < 2 $
    4. $ x<2 $
    Explanation $2^{x^{2}-x} < 2^{2} \Rightarrow x^{2}-x < 2 \Rightarrow x^{2}-x-2 < 0$.
    $(x-2)(x+1) < 0 \Rightarrow -1 < x < 2$. Correct Answer: A
16. Find $x$ if $(\frac{1}{3})^{x-1}=9$.
    1. $-1$
    2. $-2$
    3. $1$
    4. $3$
    Explanation $3^{-(x-1)} = 3^{2} \Rightarrow -x+1 = 2 \Rightarrow x = -1$. Correct Answer: A
17. The function $y=a^{x}$ passes through $(2,9)$ and $(1,b)$. Find $b$.
    1. $3$
    2. $9$
    3. $2$
    4. $\pm 3$
    Explanation $9 = a^{2} \Rightarrow a=3$ (since base $a>0$).
    $b = a^{1} = 3$. Correct Answer: A
18. If $x>0$, the minimum value of $f(x)=x+\frac{4}{x}$ is:
    1. $2$
    2. $4$
    3. $8$
    4. $0$
    Explanation Using AM-GM inequality: $x + \frac{4}{x} \ge 2\sqrt{x \cdot \frac{4}{x}} = 2\sqrt{4} = 4$. Correct Answer: B
19. Solve $9^{x}+6^{x}=2 \cdot 4^{x}$.
    1. $x=0$
    2. $x=1$
    3. $x=\log_{3/2}2$
    4. $x=-1$
    Explanation Divide by $4^{x}$: $(3/2)^{2x} + (3/2)^{x} - 2 = 0$. Let $t=(3/2)^{x}$.
    $t^{2}+t-2=0 \Rightarrow (t+2)(t-1)=0$. Since $t>0$, $t=1$.
    $(3/2)^{x}=1 \Rightarrow x=0$. Correct Answer: A
20. The set of values of $x$ for which $(x^{2}-3x+3)^{x}=1$ is:
    1. $\{0,1,2\}$
    2. $\{0,1\}$
    3. $\{1,2\}$
    4. $\{0\}$
    Explanation Cases:
    1. Exponent $x=0$: $Base \ne 0 \Rightarrow 3 \ne 0$. Valid ($x=0$).
    2. Base $= 1$: $x^{2}-3x+2=0 \Rightarrow x=1, x=2$.
    Set: $\{0, 1, 2\}$. Correct Answer: A
21. Simplify $\frac{a^{4/3}-8a^{1/3}}{a^{2/3}+2a^{1/3}+4}$.
    1. $a^{1/3}-2$
    2. $a^{2/3}-2$
    3. $a-2$
    4. $a^{2/3}(a^{1/3}-2)$
    Explanation Numerator: $a^{1/3}(a - 8) = a^{1/3}((a^{1/3})^{3} - 2^{3})$.
    Difference of cubes: $a^{1/3}(a^{1/3}-2)(a^{2/3}+2a^{1/3}+4)$.
    Denominator cancels out. Result: $a^{1/3}(a^{1/3}-2)$. Correct Answer: D
22. If $f(x)=a^{x}-a^{-x} (a>0, a \ne 1)$ is an increasing function, then:
    1. $ a > 1 $
    2. $ 0 < a < 1 $
    3. $ a > 0 $
    4. $a=1$
    Explanation $f'(x) = \ln a (a^{x} + a^{-x})$. Since $a^{x} + a^{-x} > 0$, for $f(x)$ to be increasing ($f'>0$), we need $\ln a > 0$, so $a > 1$. Correct Answer: A
23. Solve $(\sqrt{2}+1)^{x}+(\sqrt{2}-1)^{x}=6$.
    1. $x=\pm 1$
    2. $x=\pm 2$
    3. $x=2$
    4. $x=1$
    Explanation $(\sqrt{2}-1) = 1/(\sqrt{2}+1)$. Let $u=(\sqrt{2}+1)^{x}$.
    $u + 1/u = 6 \Rightarrow u^{2}-6u+1=0$. $u = 3 \pm 2\sqrt{2} = (\sqrt{2} \pm 1)^{2}$.
    If $u=(\sqrt{2}+1)^{2}$, $x=2$. If $u=(\sqrt{2}+1)^{-2}$, $x=-2$. Correct Answer: B
24. Find $m$ such that $4^{x}-m \cdot 2^{x}+m+1=0$ has real roots.
    1. $m \ge 2+2\sqrt{2}$
    2. $m \le 2-2\sqrt{2}$
    3. $m \ge 2$
    4. Both A and B
    Explanation Let $t=2^{x}$. $t^{2}-mt+m+1=0$ requires at least one positive root $t$.
    Discriminant $\ge 0 \Rightarrow m^{2}-4m-4 \ge 0$. Roots are $2 \pm 2\sqrt{2}$.
    Condition $m \ge 2+2\sqrt{2}$ ensures positive roots (Sum $>0$, Product $>0$). Correct Answer: A

Part 3: Logarithmic Problems

25. Solve $\log_{3}(x^{2}-1)-\log_{3}(x+1)=2$.
    1. $x=10$
    2. $x=8$
    3. $x=9$
    4. $x=5$
    Explanation $\log_{3}\frac{(x-1)(x+1)}{x+1} = 2 \Rightarrow \log_{3}(x-1)=2$.
    $x-1 = 3^{2}=9 \Rightarrow x=10$. Correct Answer: A
26. The domain of $y=\log_{0.5}(x^{2}-4x+3)$ is:
    1. $(1,3)$
    2. $(-\infty,1)\cup(3,+\infty)$
    3. $[1,3]$
    4. $(3,+\infty)$
    Explanation Argument $> 0 \Rightarrow x^{2}-4x+3 > 0 \Rightarrow (x-3)(x-1) > 0$.
    $x < 1$ or $x > 3$. Correct Answer: B
27. If $\log_{a}\dfrac{2}{3} < 1 $, then $a$ belongs to:
    1. $(0,\frac{2}{3})\cup(1,+\infty)$
    2. $(3,1)$
    3. $(0,3)$
    4. $(1,+\infty)$
    Explanation If$ \; a>1 , \log_{a}\dfrac{2}{3} < 1 $ holds since $ \dfrac{2}{3} < a $
    If$ \; 0 < a < 1 , \log_{a}\dfrac{2}{3} < \log_{a}a $
    $ \Rightarrow 2/3 > a \Rightarrow a < 2/3 $.
    Union: $(0, 2/3) \cup (1, +\infty)$. Correct Answer: A
28. Calculate $2 \log_{5}10 + \log_{5}0.25$.
    1. $0$
    2. $1$
    3. $2$
    4. $4$
    Explanation $\log_{5}(10^{2}) + \log_{5}(1/4) $
    $= \log_{5}(100 \cdot 0.25) = \log_{5}25 = 2$ Correct Answer: C
29. If $f(x)=\log_{a}x$ passes through $(2, 4)$, find $f(8)$.
    1. $12$
    2. $6$
    3. $3$
    4. $1.5$
    Explanation $4 = \log_{a}2 \Rightarrow a^{4}=2 \Rightarrow a=2^{1/4}$.
    $f(8) = \log_{a}8 = \log_{2^{1/4}}2^{3} = \frac{3}{1/4} = 12$. Correct Answer: A
30. Solve $\log_{2}^{2}x - 3 \log_{2}x + 2 = 0$. Product of roots is:
    1. $8$
    2. $4$
    3. $2$
    4. $3$
    Explanation $(\log_{2}x - 2)(\log_{2}x - 1) = 0$.
    $x_{1} = 2^{2} = 4, x_{2} = 2^{1} = 2$.
    Product: $4 \cdot 2 = 8$. Correct Answer: A
31. Which value is largest?
    1. $\ln 2$
    2. $\log_{2}e$
    3. $\ln 1$
    4. $\log_{0.5}2$
    Explanation $\ln 2 \approx 0.69$, $\log_{2}e \approx 1.44$, $\ln 1 = 0$, $\log_{0.5}2 = -1$.
    Largest is $\log_{2}e$. Correct Answer: B
32. Find the inverse of $f(x)=1+\log_{3}(x+2)$.
    1. $y=3^{x-1}-2$
    2. $y=3^{x+1}-2$
    3. $y=3^{x-1}+2$
    4. $y=3^{x}-3$
    Explanation $y-1 = \log_{3}(x+2) \Rightarrow 3^{y-1} = x+2 $
    $ \Rightarrow x = 3^{y-1}-2$. Correct Answer: A
33. Solve $x^{\log x}=100x$.
    1. $100, 0.1$
    2. $10, 0.1$
    3. $100, 10$
    4. $100$
    Explanation $\log(x^{\log x}) = \log(100x) $
    $ \Rightarrow (\log x)^{2} = 2 + \log x$.
    $u^{2}-u-2=0 \Rightarrow u=2, u=-1$.
    $x=10^{2}=100, x=10^{-1}=0.1$. Correct Answer: A
34. If $\lg 2=a$ and $\lg 3=b$, find $\log_{5}12$.
    1. $\frac{2a+b}{1-a}$
    2. $\frac{a+2b}{1-a}$
    3. $\frac{2a+b}{a}$
    4. $\frac{a+b}{1-a}$
    Explanation $\log_{5}12 = \dfrac{\log 12}{\log 5} $
    $= \dfrac{\log(2^{2} \cdot 3)}{\log(10/2)} $
    $= \dfrac{2\log 2 + \log 3}{1 - \log 2} = \dfrac{2a+b}{1-a}$. Correct Answer: A
35. The inequality $\log_{a}(x-1) > \log_{a}(2x+1)$ holds for $a>1$ when:
    1. No solution
    2. $x>1$
    3. $x>-1/2$
    4. $x<-2$
    Explanation $x-1 > 2x+1 \Rightarrow -x > 2 \Rightarrow x < -2$.
    Domain $x-1 > 0 \Rightarrow x > 1$.
    Intersection of $x < -2$ and $x > 1$ is Empty set. Correct Answer: A
36. Solve $\ln x + \ln(x+3)=\ln 10$.
    1. $2$
    2. $-5$
    3. $2, -5$
    4. $5$
    Explanation $x(x+3)=10 \Rightarrow x^{2}+3x-10=0 $
    $ \Rightarrow (x+5)(x-2)=0$.
    Domain $x>0$, so $x=2$. Correct Answer: A

Part 4: Trigonometry & Applications

37. Simplify $\dfrac{\sin 2x}{1+\cos 2x}$.
    1. $\tan x$
    2. $\sin x$
    3. $\cot x$
    4. $\tan 2x$
    Explanation $\dfrac{2\sin x \cos x}{2\cos^{2}x} = \tan x$. Correct Answer: A
38. The maximum value of $f(x)=\sin x+\cos x$ is:
    1. $1$
    2. $\sqrt{2}$
    3. $2$
    4. $\frac{\sqrt{2}}{2}$
    Explanation $\sqrt{1^{2}+1^{2}} = \sqrt{2}$. Correct Answer: B
39. Find the period of $y=|\sin x|$.
    1. $\pi$
    2. $2\pi$
    3. $\frac{\pi}{2}$
    4. $4\pi$
    Explanation Taking the absolute value of $\sin x$ makes the negative lobes positive, repeating every $\pi$. Correct Answer: A
40. If $\tan \alpha=2$, calculate $\dfrac{\sin \alpha - \cos \alpha}{\sin \alpha + \cos \alpha}$.
    1. $1/3$
    2. $-1/3$
    3. $3$
    4. $1$
    Explanation Divide num/den by $\cos \alpha$: $\dfrac{\tan \alpha - 1}{\tan \alpha + 1} = \dfrac{2-1}{2+1} = 1/3$. Correct Answer: A
41. Solve $\cos 2x=\sin x$ in $[0, \pi]$.
    1. $\frac{\pi}{6}, \frac{5\pi}{6}$
    2. $\frac{\pi}{6}$
    3. $\frac{\pi}{3}$
    4. $\frac{\pi}{2}$
    Explanation $1-2\sin^{2}x = \sin x \Rightarrow 2\sin^{2}x + \sin x - 1 = 0$
    $ \Rightarrow (2\sin x - 1)(\sin x + 1) = 0$.
    $\sin x = 1/2 \Rightarrow x = \pi/6, 5\pi/6$. ($\sin x = -1$ not in range). Correct Answer: A
42. The range of $y=3 \sin x+4 \cos x$ is:
    1. $[-5,5]$
    2. $[-1,1]$
    3. $[-7,7]$
    4. $[3,4]$
    Explanation Max value $\sqrt{3^{2}+4^{2}} = 5$. Range $[-5, 5]$. Correct Answer: A
43. If $\alpha, \beta \in (0,\frac{\pi}{2})$ and $\tan \alpha=\frac{1}{2}$, $\tan \beta=\frac{1}{3}$, find $\alpha+\beta$.
    1. $\frac{\pi}{4}$
    2. $\frac{\pi}{3}$
    3. $\frac{\pi}{6}$
    4. $\frac{3\pi}{4}$
    Explanation $\tan(\alpha+\beta) = \frac{1/2+1/3}{1-1/6} = \frac{5/6}{5/6} = 1$
    $ \Rightarrow \alpha+\beta = \pi/4$. Correct Answer: A
44. Graph of $y=\sin(2x-\frac{\pi}{3})$ is obtained by shifting $y=\sin 2x$ to the:
    1. Right by $\frac{\pi}{6}$
    2. Left by $\frac{\pi}{6}$
    3. Right by $\frac{\pi}{3}$
    4. Left by $\frac{\pi}{3}$
    Explanation $y = \sin(2(x-\frac{\pi}{6}))$. Shift Right by $\pi/6$. Correct Answer: A
45. Find the value of $\cos 20^{\circ} \cos 40^{\circ} \cos 80^{\circ}$.
    1. $1/8$
    2. $1/4$
    3. $1/2$
    4. $1/16$
    Explanation Multiply by $\sin 20$. Term collapses to $\frac{1}{8}\sin 160$. Since $\sin 160 = \sin 20$, result is $1/8$. Correct Answer: A
46. Which function is even?
    1. $y=x \sin x$
    2. $y=x+\sin x$
    3. $y=\cos x+\sin x$
    4. $y=x^{2} \sin x$
    Explanation $y(-x) = (-x)\sin(-x) = (-x)(-\sin x) $
    $= x\sin x = y(x)$. Correct Answer: A
47. Solve $\sqrt{3}\tan x=3$.
    1. $x=\frac{\pi}{3}+k\pi$
    2. $x=\frac{\pi}{6}+k\pi$
    3. $x=\frac{\pi}{3}+2k\pi$
    4. $x=\frac{2\pi}{3}+k\pi$
    Explanation $\tan x = 3/\sqrt{3} = \sqrt{3} \Rightarrow x = \pi/3 + k\pi$. Correct Answer: A
48. If $\sin x+\cos x=\frac{1}{2}$, find $\sin 2x$.
    1. $-3/4$
    2. $3/4$
    3. $-1/4$
    4. $-1/2$
    Explanation Squaring: $1 + \sin 2x = 1/4 \Rightarrow \sin 2x = -3/4$. Correct Answer: A