Math Problem Statement

Jawan

Diketahui f(x) = \frac{x-1}{x^2 + 3x + 2} dan g(x) = \frac{-2x + 5}{2x^2 + 3x - 5}. Domain fungsi (f+g)(x) adalah ....

Solution

We are given two functions:

f(x)=x1x2+3x+2,g(x)=2x+52x2+3x5f(x) = \frac{x - 1}{x^2 + 3x + 2}, \quad g(x) = \frac{-2x + 5}{2x^2 + 3x - 5}

The goal is to find the domain of the function (f+g)(x)(f + g)(x).

Steps to determine the domain:

  1. Identify restrictions for each function: The domain of a rational function is the set of all real numbers except those that make the denominator equal to zero.

  2. Find the values of xx that make the denominator of f(x)f(x) zero: x2+3x+2=0x^2 + 3x + 2 = 0 This quadratic can be factored: (x+1)(x+2)=0(x + 1)(x + 2) = 0 So, x=1x = -1 and x=2x = -2 make the denominator of f(x)f(x) zero, meaning f(x)f(x) is undefined for x=1x = -1 and x=2x = -2.

  3. Find the values of xx that make the denominator of g(x)g(x) zero: 2x2+3x5=02x^2 + 3x - 5 = 0 We can solve this quadratic using the quadratic formula: x=3±(3)24(2)(5)2(2)=3±9+404=3±494=3±74x = \frac{-3 \pm \sqrt{(3)^2 - 4(2)(-5)}}{2(2)} = \frac{-3 \pm \sqrt{9 + 40}}{4} = \frac{-3 \pm \sqrt{49}}{4} = \frac{-3 \pm 7}{4} So the solutions are: x=3+74=1,x=374=52x = \frac{-3 + 7}{4} = 1, \quad x = \frac{-3 - 7}{4} = -\frac{5}{2} Thus, x=1x = 1 and x=52x = -\frac{5}{2} make the denominator of g(x)g(x) zero, meaning g(x)g(x) is undefined for x=1x = 1 and x=52x = -\frac{5}{2}.

Domain of (f+g)(x)(f + g)(x):

The domain of (f+g)(x)(f + g)(x) is the set of all real numbers except the values that make either f(x)f(x) or g(x)g(x) undefined. These values are:

  • x=1x = -1
  • x=2x = -2
  • x=1x = 1
  • x=52x = -\frac{5}{2}

Thus, the domain of (f+g)(x)(f + g)(x) is all real numbers except x=1x = -1, x=2x = -2, x=1x = 1, and x=52x = -\frac{5}{2}.

Final answer:

The domain of (f+g)(x)(f + g)(x) is: R{2,1,52,1}\mathbb{R} \setminus \left\{-2, -1, -\frac{5}{2}, 1\right\}

Would you like further details on any step or have other questions?


Here are 5 related questions for practice:

  1. How would you find the domain of (fg)(x)(f - g)(x)?
  2. What is the domain of the function (f×g)(x)(f \times g)(x)?
  3. How would you calculate the domain of f(x)g(x)\frac{f(x)}{g(x)}?
  4. What happens to the domain if one of the numerators had a variable that could also become zero?
  5. How can you graphically interpret the domain of a rational function?

Tip: Always check both the numerator and denominator for restrictions when determining the domain of a rational function!

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Math Problem Analysis

Mathematical Concepts

Rational Functions
Domain
Quadratic Equations

Formulas

Quadratic equation formula: ax^2 + bx + c = 0
Rational function: f(x) = \frac{p(x)}{q(x)} where q(x) ≠ 0

Theorems

The domain of a rational function is all real numbers except those that make the denominator zero.

Suitable Grade Level

Grades 10-12