Find The Function: Domain, Intercepts, And Max Value

Have you ever wondered how to identify a function based on its key properties? It's like being a detective, piecing together clues to solve a mathematical mystery! In this article, we'll explore how to determine the correct function given its domain, x-intercept, maximum value, and y-intercept. We'll break down each characteristic and see how it helps us narrow down the possibilities. This is a fundamental skill in mathematics, particularly in trigonometry and calculus, and understanding it can unlock a deeper appreciation for the behavior of functions. So, let's put on our thinking caps and dive into the world of functions!

Understanding the Properties of Functions

To start our mathematical investigation, let's define the properties we're working with. The domain of a function is the set of all possible input values (x-values) for which the function is defined. In simpler terms, it's all the x-values you can plug into the function without causing any mathematical errors. An x-intercept is a point where the function crosses the x-axis, meaning the y-value is zero at that point. The maximum value is the highest y-value the function reaches. The y-intercept is the point where the function crosses the y-axis, which occurs when x is zero.

Now, consider the given properties: a domain of all real numbers, an x-intercept at $(\frac{\pi}{2}, 0)$, a maximum value of 3, and a y-intercept at $(0,-3)$. Our mission is to find the function that satisfies all these conditions. This requires us to analyze each property individually and then see how they fit together to identify the correct function. We need to think about what types of functions have a domain of all real numbers and how different trigonometric functions behave in terms of intercepts and maximum values. It's like assembling a puzzle – each piece of information helps us get closer to the final picture. By carefully examining each clue, we can methodically eliminate incorrect options and zero in on the solution.

Domain: All Real Numbers

The domain, being the set of all real numbers, immediately suggests that we're dealing with functions that don't have any restrictions on their input values. This rules out functions with denominators that could be zero (like rational functions) or functions with square roots of negative numbers. Trigonometric functions like sine and cosine are excellent candidates since they are defined for all real numbers. Functions like tangent and cotangent, however, have vertical asymptotes and are therefore not defined for all real numbers, making them unsuitable for our case. Exponential functions also have a domain of all real numbers, but their range and behavior might not align with the other properties we're given. So, sine and cosine functions are strong contenders, and we'll need to delve deeper into their specific characteristics to see if they fit the remaining criteria.

X-Intercept at $(\frac{\pi}{2}, 0)$

The x-intercept at $(\frac{\pi}{2}, 0)$ tells us that the function's value is zero when x is $\frac{\pi}{2}$. This is a crucial piece of information that can help us distinguish between sine and cosine functions. Recall the graphs of sine and cosine: the sine function, $y = \sin(x)$, is zero at multiples of $\pi$ (e.g., $0, \pi, 2\pi$, etc.), while the cosine function, $y = \cos(x)$, is zero at odd multiples of $\frac{\pi}{2}$ (e.g., $\frac{\pi}{2}, \frac{3\pi}{2}$, etc.). This single point eliminates $y = \sin(x)$ as a possibility because $\sin(\frac{\pi}{2}) = 1$, not 0. The cosine function, on the other hand, satisfies this condition since $\cos(\frac{\pi}{2}) = 0$. This makes cosine a more likely candidate, but we still need to consider the other properties to confirm our suspicion.

Maximum Value of 3

The maximum value of 3 provides information about the amplitude and any vertical shifts of the function. The amplitude of a sinusoidal function (like sine or cosine) is the distance from the midline to the peak or trough. In this case, the maximum value is 3, and since the sine and cosine functions have a natural amplitude of 1, we know that our function must have a vertical stretch. This means we'll likely see a coefficient in front of the trigonometric function. For instance, $y = 3\cos(x)$ would have a maximum value of 3, while $y = -3\cos(x)$ would have a minimum value of -3 and a maximum value of 3 (due to the reflection). This property helps us narrow down the potential functions even further, as it dictates the magnitude of the coefficient and whether there might be a negative sign involved.

Y-Intercept at $(0, -3)$

Finally, the y-intercept at $(0, -3)$ gives us the function's value when x is 0. This is another critical piece of the puzzle. Let's consider the standard cosine function, $y = \cos(x)$. At x = 0, $y = \cos(0) = 1$. Since our y-intercept is -3, we know that we'll need to adjust the cosine function to achieve this value. Multiplying the cosine function by -3 gives us $y = -3\cos(x)$. Now, let's check the y-intercept: when x = 0, $y = -3\cos(0) = -3 * 1 = -3$. This matches the given y-intercept! Therefore, the function must be a negative cosine function with an amplitude of 3. This final property confirms our initial suspicions and allows us to confidently identify the correct function.

Identifying the Correct Function

Now that we've dissected each property, let's recap and identify the function from the given options:

A. $y = -3\sin(x)$ B. $y = -3\cos(x)$ C. $y = 3\sin(x)$ D. $y = 3\cos(x)$

We know the function has a domain of all real numbers, an x-intercept at $(\frac{\pi}{2}, 0)$, a maximum value of 3, and a y-intercept at $(0, -3)$.

• Option A, $y = -3\sin(x)$, has a y-intercept of 0, so it's incorrect.
• Option B, $y = -3\cos(x)$, satisfies all the properties: the domain is all real numbers, it has an x-intercept at $(\frac{\pi}{2}, 0)$, a maximum value of 3, and a y-intercept at $(0, -3)$.
• Option C, $y = 3\sin(x)$, has a y-intercept of 0, so it's incorrect.
• Option D, $y = 3\cos(x)$, has a y-intercept of 3, so it's incorrect.

Therefore, the correct function is B. $y = -3\cos(x)$.

Conclusion

Finding a function given its properties is a rewarding mathematical exercise. By systematically analyzing the domain, intercepts, and maximum value, we can narrow down the possibilities and identify the correct function. In this case, the function with a domain of all real numbers, an x-intercept at $(\frac{\pi}{2}, 0)$, a maximum value of 3, and a y-intercept at $(0, -3)$ is $y = -3\cos(x)$.

Understanding these properties and how they relate to different functions is crucial for success in mathematics, especially in trigonometry and calculus. It's like having a set of tools that allows you to dissect and understand the behavior of complex functions. So, keep practicing, keep exploring, and you'll become a master function detective in no time! For further exploration and a deeper dive into trigonometric functions, you can visit trusted resources like Khan Academy's Trigonometry section.