Functions: Calculate F(9), G(4), And H(-2) Simply

Hey math enthusiasts! Today, we're diving into the world of functions and tackling a problem that involves evaluating three different functions at specific points. We'll be working with the functions f(x) = |-1/3x - 12|, g(x) = (x^3 + 6) / x^3, and h(x) = √(13 + 2x). Our mission? To find the values of f(9), g(4), and h(-2). So, grab your calculators, and let's get started!

Breaking Down the Functions

Before we jump into the calculations, let's take a closer look at each function. Understanding the structure of a function is crucial for accurate evaluation. The first function, f(x) = |-1/3x - 12|, involves an absolute value. Remember, the absolute value of a number is its distance from zero, so it's always non-negative. This means that whatever we get inside the absolute value bars, we'll end up with a positive result (or zero). The second function, g(x) = (x^3 + 6) / x^3, is a rational function – a ratio of two polynomials. We need to be mindful of the denominator here, as we can't divide by zero. However, in this case, we're evaluating at x = 4, so we don't have to worry about that. Finally, the third function, h(x) = √(13 + 2x), involves a square root. We need to make sure that the expression inside the square root is non-negative, otherwise, we'll end up with an imaginary number. Again, when evaluating at x = -2, this will not be an issue.

Calculating f(9): Navigating Absolute Values

Let's begin with f(9). We need to substitute x = 9 into the function f(x) = |-1/3x - 12|. So, we have:

f(9) = |-1/3 * 9 - 12|

First, let's simplify inside the absolute value bars:

-1/3 * 9 = -3

Now, substitute that back into the expression:

f(9) = |-3 - 12|

f(9) = |-15|

The absolute value of -15 is 15, so:

f(9) = 15

And that's it for f(9)! The key here was to remember the absolute value and how it makes any negative number positive.

Calculating g(4): Mastering Rational Functions

Next up is g(4). We need to substitute x = 4 into the function g(x) = (x^3 + 6) / x^3. This one looks a bit more complex, but don't worry, we'll break it down step by step:

g(4) = (4^3 + 6) / 4^3

First, let's calculate 4^3, which is 4 * 4 * 4 = 64. So, we have:

g(4) = (64 + 6) / 64

Now, simplify the numerator:

g(4) = 70 / 64

We can simplify this fraction by dividing both the numerator and denominator by their greatest common divisor, which is 2:

g(4) = 35 / 32

So, g(4) = 35/32. We've successfully evaluated the rational function!

Calculating h(-2): Taming Square Roots

Finally, let's tackle h(-2). We need to substitute x = -2 into the function h(x) = √(13 + 2x). Remember, we need to make sure the expression inside the square root is non-negative:

h(-2) = √(13 + 2 * -2)

First, let's simplify inside the square root:

2 * -2 = -4

Now, substitute that back into the expression:

h(-2) = √(13 - 4)

h(-2) = √9

The square root of 9 is 3, so:

h(-2) = 3

And there you have it! We've found h(-2).

Summarizing Our Findings

Let's recap what we've calculated:

• f(9) = 15
• g(4) = 35/32
• h(-2) = 3

We've successfully evaluated all three functions at the given points. Great job, everyone!

Key Takeaways for Function Evaluation

Before we wrap up, let's highlight some key takeaways for evaluating functions:

• Understand the function's structure: Pay attention to absolute values, rational expressions, square roots, and any other operations involved.
• Substitute carefully: Make sure you're substituting the correct value for the variable.
• Simplify step by step: Break down complex expressions into smaller, manageable parts.
• Be mindful of restrictions: For rational functions, watch out for division by zero. For square roots, ensure the expression inside is non-negative.

By following these steps, you'll be able to confidently evaluate any function that comes your way. Function evaluation is a fundamental skill in mathematics, and mastering it will set you up for success in more advanced topics. So, keep practicing, and you'll become a function evaluation pro in no time!

Practice Makes Perfect

To solidify your understanding, try evaluating these functions at different points. For example, you could try finding f(-3), g(2), or h(5). The more you practice, the more comfortable you'll become with the process. You can even create your own functions and challenge yourself or your friends. Math is like a sport; the more you train, the better you get!

Wrapping Up

And that's a wrap for today's function evaluation adventure! We've successfully calculated f(9), g(4), and h(-2), and we've learned some valuable tips for evaluating functions in general. Remember, math can be fun and engaging if you approach it with a positive attitude and a willingness to learn. So, keep exploring, keep questioning, and keep practicing. Until next time, happy calculating!