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Notely Maths 12
Class 12 Maths
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1.1 Q-4

Question Statement

Find the domain and the range of the following functions and sketch their graphs:

i. g(x)=2xβˆ’5g(x) = 2x - 5

ii. g(x)=x2βˆ’4g(x) = \sqrt{x^2 - 4}

iii. g(x)=x+1g(x) = \sqrt{x + 1}

iv. g(x)=∣xβˆ’3∣g(x) = |x - 3|

v. g(x)={6x+7,xβ‰€βˆ’2xβˆ’3,βˆ’2<xg(x) = \begin{cases} 6x + 7, & x \leq -2 x - 3, & -2 < x \end{cases}
vi. g(x)={xβˆ’1,x<32x+1,3≀xg(x) = \begin{cases} x - 1, & x < 3 2x + 1, & 3 \leq x \end{cases} vii. g(x)=x2βˆ’3x+2x+1,xβ‰ βˆ’1g(x) = \frac{x^2 - 3x + 2}{x + 1}, x \neq -1

viii. g(x)=x2βˆ’16xβˆ’4,xβ‰ 4g(x) = \frac{x^2 - 16}{x - 4}, x \neq 4

Background and Explanation

To solve these types of questions, you need to understand the concepts of domain and range:

  • Domain: The set of all possible input values (x-values) for which the function is defined.
  • Range: The set of all possible output values (y-values) the function can take based on its domain.

Additionally, sketching the graph of a function helps visualize its behavior, providing insights into how the function changes across its domain.

Solution

i. g(x)=2xβˆ’5g(x) = 2x - 5

  • Domain: The function is a linear function, so it is defined for all real values of xx.
    • Domain: (βˆ’βˆž,∞)(-\infty, \infty)
  • Range: A linear function produces all real values for yy, so the range is also all real numbers.
    • Range: (βˆ’βˆž,∞)(-\infty, \infty) Graph:
      Pasted image 20241206055721.png

ii. g(x)=x2βˆ’4g(x) = \sqrt{x^2 - 4}

  • Domain: The expression inside the square root must be non-negative, i.e., x2βˆ’4β‰₯0x^2 - 4 \geq 0.
    Solving this gives xβ‰€βˆ’2x \leq -2 or xβ‰₯2x \geq 2.
    • Domain: (βˆ’βˆž,βˆ’2]βˆͺ[2,∞)(-\infty, -2] \cup [2, \infty)
  • Range: Since the square root produces non-negative values, the range is all non-negative real numbers.
    • Range: [0,∞)[0, \infty) Graph:
      Pasted image 20241206055758.png

iii. g(x)=x+1g(x) = \sqrt{x + 1}

  • Domain: The expression inside the square root must be non-negative, i.e., x+1β‰₯0x + 1 \geq 0.
    Solving this gives xβ‰₯βˆ’1x \geq -1.
    • Domain: [βˆ’1,∞)[-1, \infty)
  • Range: Since the square root produces non-negative values, the range is all non-negative real numbers.
    • Range: [0,∞)[0, \infty) Graph:
      Pasted image 20241206055827.png

iv. g(x)=∣xβˆ’3∣g(x) = |x - 3|

  • Domain: Absolute value functions are defined for all real xx values.
    • Domain: (βˆ’βˆž,∞)(-\infty, \infty)
  • Range: Since the absolute value is always non-negative, the range is all non-negative real numbers.
    • Range: [0,∞)[0, \infty) Graph:
      Pasted image 20241206055932.png

v. g(x)={6x+7,xβ‰€βˆ’2xβˆ’3,βˆ’2<xg(x) = \begin{cases} 6x + 7, & x \leq -2 x - 3, & -2 < x \end{cases}

  • Domain: The piecewise function is defined for all real numbers, excluding any gaps.
    • Domain: (βˆ’βˆž,∞)(-\infty, \infty)
  • Range: Since both parts of the function can take all real values, the range is all real numbers.
    • Range: (βˆ’βˆž,∞)(-\infty, \infty) Graph:
      Pasted image 20241206060034.png

vi. g(x)={xβˆ’1,x<32x+1,xβ‰₯3g(x) = \begin{cases} x - 1, & x < 3 2x + 1, & x \geq 3 \end{cases}

  • Domain: The piecewise function is defined for all real values of xx.
    • Domain: (βˆ’βˆž,∞)(-\infty, \infty)
  • Range: Since both parts of the function can take all real values, the range is all real numbers.
    • Range: (βˆ’βˆž,∞)(-\infty, \infty) Graph:
      Pasted image 20241206060123.png

vii. g(x)=x2βˆ’3x+2x+1,xβ‰ βˆ’1g(x) = \frac{x^2 - 3x + 2}{x + 1}, x \neq -1

  • Domain: The denominator cannot be zero, so xβ‰ βˆ’1x \neq -1.

    • Domain: (βˆ’βˆž,βˆ’1)βˆͺ(βˆ’1,∞)(-\infty, -1) \cup (-1, \infty)

  • Range: Since the simplified form of the function is g(x)=x+2g(x) = x + 2, the range is all real values except x=βˆ’1x = -1.

    • Range: (βˆ’βˆž,∞)(-\infty, \infty)

    Graph:
    Pasted image 20241206060207.png

viii. g(x)=x2βˆ’16xβˆ’4,xβ‰ 4g(x) = \frac{x^2 - 16}{x - 4}, x \neq 4

  • Domain: The denominator cannot be zero, so xβ‰ 4x \neq 4.
    • Domain: (βˆ’βˆž,4)βˆͺ(4,∞)(-\infty, 4) \cup (4, \infty)
  • Range: Since the function can take all real values except at x=4x = 4, the range is all real numbers except x=4x = 4.
    • Range: (βˆ’βˆž,∞)(-\infty, \infty) Graph:
      Pasted image 20241206060247.png