Logarithmic Function Domain: $f(x)=\ln \left(-x^2+49\right)$

Hey everyone! Today, we're diving deep into the fascinating world of logarithmic functions and, more specifically, figuring out their domain. You know, that crucial set of input values (the 'x's) that a function can actually accept without throwing a mathematical tantrum. Our specific mission, should we choose to accept it, is to analytically determine the domain of this beast: $f(x)=\ln \left(-x^2+49\right)$. Get ready to flex those analytical muscles, guys, because we're going to break this down step-by-step.

Understanding the Logarithm's Requirements

Before we even touch our specific function, let's get a grip on what logs need. The logarithmic function, denoted as $\ln(u)$ or $\log(u)$, has a very strict rule: the argument must be greater than zero. That's it. No negatives, no zeros. Think of it like a bouncer at an exclusive club; only positive numbers get in. So, for our function $f(x)=\ln \left(-x^2+49\right)$, the entire expression inside the logarithm, which is $-x^2+49$, has to be strictly positive. This is the fundamental principle that will guide our entire analysis. We're not just randomly picking numbers; we're using the inherent properties of logarithms to define the boundaries of our domain. This isn't just about finding a range of numbers; it's about understanding why that range exists based on the mathematical definition of the logarithm itself. We need to ensure that whatever 'x' value we plug in, the result of $-x^2+49$ is something that the natural logarithm function can actually process. If it's zero or negative, the logarithm is undefined, and therefore, the function $f(x)$ is undefined for that particular 'x'. So, our primary task is to translate this requirement into a solvable inequality.

Setting Up the Inequality

Alright, armed with the knowledge that the argument of a logarithm must be positive, we can now translate this into a concrete mathematical inequality for our function. The argument of the natural logarithm in f(x)=\ln \left(-x^2+49 ight) is the expression $-x^2+49$. Therefore, to find the domain, we need to solve the following inequality:

$-x^2 + 49 > 0$

This inequality is the core of our problem. It directly represents the condition for which our logarithmic function is defined. We need to find all the values of 'x' that satisfy this condition. This isn't just a simple substitution; it's about building a mathematical model of the function's limitations. Think of it as reverse-engineering the function's behavior. By ensuring the argument is positive, we guarantee that the output of the function is a real number. This inequality will be the foundation upon which we build our solution. It's the gatekeeper, and we need to figure out which 'x' values allow passage. The goal is to isolate 'x' or understand the range of 'x' values that make this statement true. This process involves algebraic manipulation, and we need to be careful with our steps, especially when dealing with quadratic expressions and inequalities. This inequality is the key; everything else flows from solving it correctly.

Solving the Quadratic Inequality

Now comes the fun part, guys: solving the quadratic inequality $-x^2 + 49 > 0$. This is where we get to do some good old-fashioned algebra. A common strategy for solving quadratic inequalities is to first find the roots of the corresponding quadratic equation, which is $-x^2 + 49 = 0$. Let's do that:

$-x^2 = -49$

Multiply both sides by -1:

$x^2 = 49$

Now, take the square root of both sides:

$x =

±

\sqrt{49}$

$x =

±

7$

So, the roots are $x = 7$ and $x = -7$. These roots are critical because they are the points where the expression $-x^2 + 49$ equals zero. For a quadratic function, these roots divide the number line into intervals. We need to determine in which of these intervals the expression $-x^2 + 49$ is greater than zero.

The roots $-7$ and $7$ divide the number line into three intervals: $(-\infty, -7)$, $(-7, 7)$, and $(7,

\infty)$.

To figure out which interval(s) satisfy our inequality, we can use a test value from each interval. Let's pick a value within each interval and plug it back into our inequality $-x^2 + 49 > 0$:

1. Interval $(-\infty, -7)$: Let's choose $x = -8$. Plugging this into the inequality: $-(-8)^2 + 49 = -(64) + 49 = -15$. Is $-15 > 0$? No. So, this interval is not part of our domain.

2. Interval $(-7, 7)$: Let's choose $x = 0$. Plugging this into the inequality: $-(0)^2 + 49 = 0 + 49 = 49$. Is $49 > 0$? Yes. So, this interval is part of our domain.

3. **Interval $(7,

\infty)$:** Let's choose $x = 8$. Plugging this into the inequality: $-(8)^2 + 49 = -(64) + 49 = -15$. Is $-15 > 0$? No. So, this interval is not part of our domain.

Alternatively, we can think about the graph of the quadratic $y = -x^2 + 49$. This is a parabola that opens downwards (because of the negative coefficient of $x^2$) and has its vertex at $(0, 49)$. The 'x'-intercepts are at $-7$ and $7$. Since we are looking for where $-x^2 + 49 > 0$, we are looking for the 'x' values where the parabola is above the x-axis. This occurs precisely between the two x-intercepts. So, the inequality is satisfied for all 'x' values strictly between $-7$ and $7$.

Stating the Domain

Based on our analysis of the quadratic inequality, we found that the expression $-x^2 + 49$ is greater than zero for all values of 'x' strictly between $-7$ and $7$. This means that the input values for which our logarithmic function f(x)=\ln \left(-x^2+49 ight) is defined are exactly these values. Therefore, the domain of the function is the open interval $(-7, 7)$.

We express this in interval notation as $(-7, 7)$. In set-builder notation, it would be { $x$ | $-7 < x < 7$ }. This means that any number you choose between $-7$ and $7$ (but not including $-7$ or $7$ themselves) will produce a valid output for our function. For example, if you plug in $x=0$, you get $f(0) = \ln(49)$, which is a real number. If you try to plug in $x=7$ or $x=-7$, you get $\ln(0)$, which is undefined. If you try to plug in $x=8$, you get $\ln(-15)$, which is also undefined. So, the interval $(-7, 7)$ is indeed the correct domain.

This analytical approach ensures that we haven't missed any possibilities and that our domain is precisely defined according to the rules of logarithms and inequalities. It's all about respecting the function's inherent mathematical constraints. The domain is the set of all possible 'valid' inputs, and we've meticulously identified that set for f(x)=\ln \left(-x^2+49 ight) by ensuring the argument of the logarithm remains strictly positive. This process is fundamental in understanding function behavior and is applicable to many other types of functions as well. Keep practicing these analytical steps, and you'll become a domain-finding pro in no time!