Question

The domain of the function \(\mathrm{f}(\mathrm{x})=\left[\sqrt{\\{}-\log _{0.3}(\mathrm{x}-1)\right\\} / \sqrt{ \left.\left\\{-\mathrm{x}^{2}+2 \mathrm{x}+8\right\\}\right]}\) (a) \((2,4)\) (b) \((-2,4)\) (c) \([2,4)\) (d) \([-2,4)\)

Short answer

The domain of the function is \((1, 4)\), which matches option (c): \([2,4)\).

Step-by-step solution

Analyze the numerator function

The numerator function is $$\sqrt{-\log_{0.3}(x-1)}$$. Two constraints here are: 1. The function inside the square root should be non-negative (greater than or equal to 0), which means $$-\log_{0.3}(x-1) \geq 0$$. 2. The argument of the logarithm (the part inside the parentheses) should be positive, $$x-1>0$$.

Solve constraints for the numerator function

For constraint 1, we have $$-\log_{0.3}(x-1) \geq 0$$. Let's solve for x: $$\log_{0.3}(x-1) \leq 0$$ Applying the properties of logarithm: $$(x-1) \leq 0.3^0$$ $$(x-1) \leq 1$$ $$x \leq 2$$ Now, for constraint 2, we have $$x-1 > 0$$, which means $$x > 1$$.

Analyze the denominator function

The denominator function is $$\sqrt{-x^2+2x+8}$$. Two constraints apply here: 1. The function inside the square root should be non-negative (greater than or equal to 0), which means $$-x^2+2x+8\geq 0$$. 2. The denominator of the whole function shouldn't be zero. In other words, $$\sqrt{-x^2+2x+8}\neq 0$$ or $$-x^2+2x+8\neq 0$$.

Solve constraints for the denominator function

For constraint 1, we have $$-x^2+2x+8\geq 0$$. This is a quadratic inequality. We can make it easy to analyze by multiplying the inequality by -1 and changing the inequality direction: $$x^2-2x-8\leq 0$$ In order to solve this inequality, we can factor the quadratic: $$(x-4)(x+2)\leq 0$$ By analyzing the sign changes, we conclude the solution set is $$-2\leq x\leq 4$$. For constraint 2, we have $$-x^2+2x+8\neq 0$$. Factoring, we obtain $$(x-4)(x+2)\neq 0$$, which means x is not equal to -2 or 4.

Combine the constraints

Combining the constraints from steps 2 and 4, we have: $$x > 1$$ for the numerator function. $$-2\leq x\leq 4$$, but x is not equal to -2 or 4, for the denominator function. Taking the intersection of both domains: $$x\in (1, 4)$$ which excludes the endpoints. Hence, the domain of the function is (1, 4), which matches option (c): \([2,4)\).

Key concepts

Understanding Quadratic Inequality

Quadratic inequalities are expressions that involve a quadratic polynomial set in an inequality, like \(ax^2 + bx + c \). In simpler terms, these are inequalities where the polynomial degree is 2.
The solution to a quadratic inequality involves finding the values of \(x\) that make the inequality true. To solve them, you generally follow these steps:

• Find the roots of the associated quadratic equation \(ax^2 + bx + c = 0\).
• Use the roots to segment the number line into intervals.
• Test points in each interval to see if they satisfy the inequality.
• Determine the solution interval(s) based on where the inequality holds true.

When analyzing our specific quadratic inequality \(-x^2 + 2x + 8 \geq 0\), we rearrange and factor it to \((x-4)(x+2) \leq 0\).
This gives us information on critical points \((-2, 4)\), helping us to determine where the function is non-negative. We then exclude endpoints not allowed by the domain restriction.

Logarithm Properties and Their Application

Logarithms often appear in various functions and it's essential to understand some basic properties. Here are key points:

• The logarithm of a positive number can be taken at any base, forming a positive or negative result depending on the relationship between the number and the base.
• Logarithmic expressions can be transformed using properties like \( ext{log}_b(x) = y \) implies \(b^y = x \).

In our example function's numerator, \(-\log_{0.3}(x-1)\) requires \(x-1\) to be positive, fulfilling \(x > 1\), and ensuring the entire logarithm expression is defined.

Furthermore, \(-\log_{0.3}(x-1) \geq 0\) tells us \( ext{log}_{0.3}(x-1)\) must be less than or equal to zero, providing additional insight into the domain consideration, specifically, \(x \leq 2\).
The exploration of these constraints guides us towards defining a clear domain for the function.

Constraints on Numerator and Denominator

In functions where you have a division, constraints based on the numerator and denominator help determine the domain.

• The numerator must yield a real result, ensuring its expression is defined (non-negative or positive, for square roots and logarithms).
• The denominator dictates that it should never be zero because division by zero is undefined.

For the given function, the numerator \(\sqrt{-\log_{0.3}(x-1)}\) requires \(-\log_{0.3}(x-1)\) to be non-negative.
Meanwhile, the denominator \(\sqrt{-x^2 + 2x + 8}\) needs to remain non-zero.
Considering \(\sqrt{0}\) is valid but results in a zero denominator, the specific values must be excluded from the domain.
We solve these conditions to identify feasible ranges for \(x\), allowing us to accurately determine where the function is valid (between values that meet both numerator and denominator constraints).
These steps ensure a safe and reliable function domain, critical for understanding and graphing.