Question

Domain of definition of the function \(f(x)=\left[3 /\left(4-x^{2}\right)\right]+\log _{10}\left(x^{3}-x\right)\) is (a) \((-1,0) \cup(1,2) \cup(2, \infty)\) (b) \((-1,0) \cup(1,0)\) (c) \((-2,2)\) (d) \((1,2) \cup(2, \infty)\)

Short answer

The domain of definition of the function \(f(x)=\left[\frac{3}{(4-x^{2})}\right]+\log_{10}(x^{3}-x)\) is \((-1, 0) \cup (1, 2) \cup (2, \infty)\) (option a).

Step-by-step solution

Analyze the denominator

The first part of the function has a denominator \((4 - x^2)\), which we need to ensure is never zero. That means the domain of the function must avoid values of x that make this denominator zero: \(4 - x^2 \neq 0\) Solving for x, we have: \(x^2 \neq 4\) So, \(x \neq \pm 2\).

Analyze the logarithmic argument

The second part of the function has a logarithmic function with the argument \((x^3 - x)\). We need to ensure that the argument is always positive, i.e.,: \(x^3 - x > 0\) We can factor the expression as: \(x(x^2 - 1) > 0\) This can be further factored as: \(x(x - 1)(x + 1) > 0\) Now, we need to analyze the intervals determined by the zeros of this expression: x = -1, 0, and 1. 1. For x in \((- \infty, -1)\), all three factors are negative, so the product is negative. 2. For x in \((-1, 0)\), the factors (x - 1) and (x + 1) are negative, and x is positive, so the product is positive. 3. For x in \((0, 1)\), x is positive, but the factors (x - 1) and (x + 1) are negative, so the product is negative. 4. For x in \((1, \infty)\), all three factors are positive, so the product is positive. Therefore, the domain of the logarithmic argument is \((-1, 0) \cup (1, \infty)\).

Combine the domains

Now, we need to combine the domains obtained in steps 1 and 2, remembering that x cannot be equal to \(\pm 2\) and the domain of the logarithmic argument is \((-1, 0) \cup (1, \infty)\): \((-1, 0) \cup (1, 2) \cup (2, \infty)\) So, the correct domain of definition of the function is given by option (a) \((-1, 0) \cup (1, 2) \cup (2, \infty)\).

Key concepts

Logarithmic Functions

Logarithmic functions are intriguing, not least because they are the inverse of exponential functions. A basic understanding of logarithms is crucial in many areas of mathematics, particularly when dealing with exponential growth and decay problems.

When looking at the function given in the exercise, \(f(x) = \left[3 / \left(4 - x^{2}\right)\right] + \log_{10}\left(x^{3} - x\right)\), the term \(\log_{10}\left(x^{3} - x\right)\) represents the logarithmic component of the function. This tells us that we're working with a base-10 logarithm.

The key rule to remember with logarithms is that the argument—what you find inside the log function—must always be greater than zero. This is because you can't take the logarithm of a non-positive number. Hence, establishing the domain of the logarithm itself necessitates solving inequalities to ensure all values within that domain meet the requirement of having a positive argument. The resulting values of x that meet this condition help to determine the overall domain of the function.

Inequalities

Inequalities are mathematical expressions that describe the relative size or order of two values. They're an essential part of determining the domain of a function, as they help identify which values a variable can take.

For the function in our exercise, the inequality comes into play when determining the domain of the logarithmic part; \(x^3 - x > 0\). Inequalities come with their own rules, like how multiplying or dividing both sides by a negative number reverses the inequality sign, and these rules are vital during the solving process.

When we factorize \(x^3 - x\), we get \(x(x^2 - 1) > 0\), which further simplifies to \(x(x - 1)(x + 1) > 0\). To find the intervals where this inequality is true, we test values from the different intervals created by the zeros of the expression, which boils down to interval analysis. The result from this gives us segments on the x-axis where our inequalities hold true, thus contributing to defining the domain of the entire function.

Factorization

Factorization is a method of breaking down algebraic expressions into simpler parts, or 'factors,' that, when multiplied together, give back the original expression. Factorization is especially useful in solving polynomial inequalities as well as finding zeroes of polynomial functions.

The exercise involved factorizing the expression \(x^3 - x\) to simplify the inequality. This was accomplished by recognizing \(x^3 - x\) as a difference of cubes and breaking it down into \(x(x^2 - 1)\), which then factored further into \(x(x - 1)(x + 1)\). Proper factorization allows for a clearer understanding of when individual components of the product are positive or negative, which, in turn, helps to establish the intervals where the original inequality holds true. Mastering factorization is essential for those aiming to excel in algebra and beyond.