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Find the domain of existence for each of the following functions: a) 10x5 - 2x + (1)/(3) b) (x)/(x2 - 4) c) [4]x-10

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4 parts
a

R\mathbb{R}

b

R∖{−2,2}\mathbb{R} \setminus \{-2, 2\}

c

[10,+∞)[10, +\infty)

d

[−1,+∞)[-1, +\infty)

Step-by-step solution

14 steps

Here are the domains of existence for each expression:

Part a· 2 steps

10x5−2x+1310x^5 - 2x + \frac{1}{3}

  1. Classify the expression.

    This is a polynomial function.

  2. Determine the domain of existence.

    Polynomial functions are defined for all real numbers.

    D=(−∞,+∞)D = (-\infty, +\infty)

    The domain is R\boxed{\mathbb{R}}.

Part b

R∖{−2,2}\mathbb{R} \setminus \{-2, 2\}

xx2−4\frac{x}{x^2 - 4}

  1. Classify the expression.

    This is a rational function.

  2. Determine the conditions for existence.

    For a rational function, the denominator cannot be zero.

    x2−4≠0x^2 - 4 \neq 0

  3. Solve for xx where the denominator is zero.

    x2−4=0x^2 - 4 = 0 (x−2)(x+2)=0(x-2)(x+2) = 0 x−2=0orx+2=0x-2 = 0 \quad \text{or} \quad x+2 = 0 x=2orx=−2x = 2 \quad \text{or} \quad x = -2

  4. State the domain.

    The function is defined for all real numbers except x=2x=2 and x=−2x=-2.

    D={x∈R∣x≠−2 and x≠2}D = \{x \in \mathbb{R} \mid x \neq -2 \text{ and } x \neq 2\}

    In interval notation:

    D=(−∞,−2)∪(−2,2)∪(2,+∞)D = (-\infty, -2) \cup (-2, 2) \cup (2, +\infty)

    The domain is R∖{−2,2}\boxed{\mathbb{R} \setminus \{-2, 2\}}.

Part c

[10,+∞)[10, +\infty)

x−104\sqrt[4]{x-10}

  1. Classify the expression.

    This is a radical function with an even index (fourth root).

  2. Determine the conditions for existence.

    For an even root, the expression under the radical must be non-negative.

    x−10≥0x-10 \ge 0

  3. Solve the inequality.

    x≥10x \ge 10

  4. State the domain.

    D={x∈R∣x≥10}D = \{x \in \mathbb{R} \mid x \ge 10\}

    In interval notation:

    D=[10,+∞)D = [10, +\infty)

    The domain is [10,+∞)\boxed{[10, +\infty)}.

Part d

[−1,+∞)[-1, +\infty)

x+1\sqrt{x+1}

  1. Classify the expression.

    This is a radical function with an even index (square root).

  2. Determine the conditions for existence.

    For an even root, the expression under the radical must be non-negative.

    x+1≥0x+1 \ge 0

  3. Solve the inequality.

    x≥−1x \ge -1

  4. State the domain.

    D={x∈R∣x≥−1}D = \{x \in \mathbb{R} \mid x \ge -1\}

    In interval notation:

    D=[−1,+∞)D = [-1, +\infty)

    The domain is [−1,+∞)\boxed{[-1, +\infty)}.

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