Function Evaluation And Solution: G(x) = 2x^2 - 4

Hey guys! Let's dive into a fun math problem involving functions. We're given a function, and we need to figure out a couple of things: first, what happens when we plug in specific numbers, and second, what numbers give us a specific output. This is a classic algebra exercise, and we'll break it down step by step. So, let’s get started!

Understanding the Function g(x)

Before we jump into solving, let's make sure we understand what the function g(x) = 2x^2 - 4 actually means. In simple terms, this function takes an input value (which we call x), does a couple of things to it, and then spits out a result. Specifically, it:

1. Squares the input value (x).
2. Multiplies the result by 2.
3. Subtracts 4 from the product.

Think of it like a little machine where you feed in a number, and it churns it through these operations to give you a final answer. Understanding this process is crucial for both evaluating the function at specific points and solving for x when we know the output.

Now, let’s delve into the first part of our problem: finding the value of g(4) + g(-3). This essentially means we need to run the function machine twice, once with x = 4 and once with x = -3, and then add the results together. This type of problem highlights the importance of order of operations (PEMDAS/BODMAS) and careful substitution, so let's take our time and get it right.

Part a: Evaluating g(4) + g(-3)

This part asks us to find the sum of $g(4)$ and $g(-3)$. Essentially, we need to plug in $4$ and $-3$ into the function $g(x) = 2x^2 - 4$ and then add the results.

Step 1: Evaluate g(4)

To find $g(4)$, we substitute $x$ with $4$ in the function:

$g(4) = 2(4)^2 - 4$

Following the order of operations (PEMDAS/BODMAS), we first square the $4$:

$g(4) = 2(16) - 4$

Next, we multiply by $2$:

$g(4) = 32 - 4$

Finally, we subtract $4$:

$g(4) = 28$

So, $g(4)$ equals $28$.

Step 2: Evaluate g(-3)

Now, let's find $g(-3)$. We substitute $x$ with $-3$ in the function:

$g(-3) = 2(-3)^2 - 4$

Again, we start by squaring the $-3$. Remember that a negative number squared becomes positive:

$g(-3) = 2(9) - 4$

Multiply by $2$:

$g(-3) = 18 - 4$

Subtract $4$:

$g(-3) = 14$

Therefore, $g(-3)$ is $14$.

Step 3: Calculate g(4) + g(-3)

Now that we have both $g(4)$ and $g(-3)$, we can add them together:

$g(4) + g(-3) = 28 + 14$

$g(4) + g(-3) = 42$

So, the final answer for part a is 42. We successfully evaluated the function at two different points and found their sum. This highlights the direct application of the function definition and the importance of careful arithmetic.

Now, let’s tackle the second part of the problem, which involves a slightly different challenge: finding the value(s) of x that make the function equal to a specific output. This requires us to solve an equation, a fundamental skill in algebra.

Part b: Finding x When g(x) = 46

In this part, we're asked to find the value(s) of $x$ for which $g(x) = 46$. This means we need to solve the equation:

$2x^2 - 4 = 46$

This is a quadratic equation, and our goal is to isolate $x$.

Step 1: Isolate the term with x^2

First, we add $4$ to both sides of the equation:

$2x^2 - 4 + 4 = 46 + 4$

$2x^2 = 50$

Step 2: Isolate x^2

Next, we divide both sides by $2$:

rac{2x^2}{2} = rac{50}{2}

$x^2 = 25$

Step 3: Solve for x

Now, we need to find the values of $x$ that, when squared, equal $25$. Remember that both positive and negative numbers can result in a positive square. So, we take the square root of both sides:

$\sqrt{x^2} = \pm\sqrt{25}$

$x = \pm 5$

This gives us two solutions: $x = 5$ and $x = -5$.

Step 4: Verify the Solutions (Optional but Recommended)

It's always a good idea to check our answers by plugging them back into the original equation:

• For $x = 5$:

  $g(5) = 2(5)^2 - 4 = 2(25) - 4 = 50 - 4 = 46$ (Correct!)

• For $x = -5$:

  $g(-5) = 2(-5)^2 - 4 = 2(25) - 4 = 50 - 4 = 46$ (Correct!)

Both solutions satisfy the equation, so we're confident in our answer. The values of $x$ that make $g(x) = 46$ are 5 and -5. This emphasizes the importance of considering both positive and negative roots when solving equations involving squares.

Key Takeaways

Alright guys, we've successfully navigated this function problem! Let's recap what we've learned:

• Function Evaluation: We practiced substituting values into a function and simplifying to find the output. Remember to follow the order of operations (PEMDAS/BODMAS) carefully.
• Solving for x: We learned how to solve for x when given the output of a function. This often involves isolating x and using inverse operations (like taking the square root).
• Quadratic Equations: We encountered a quadratic equation ($x^2 = 25$) and remembered that it can have two solutions (both positive and negative roots).
• Verification: We highlighted the importance of checking your answers by plugging them back into the original equation. This helps prevent errors and builds confidence in your solutions.

These are fundamental concepts in algebra and are crucial for tackling more complex problems later on. So, make sure you're comfortable with these techniques!

Practice Makes Perfect

To solidify your understanding, try working through similar problems. You can change the function, the input values, or the desired output to create new challenges. For example:

• What is $g(2) + g(-1)$?
• Find $x$ if $g(x) = 14$.
• Consider a new function, $h(x) = x^3 + 2x$. Find $h(3)$ and solve for $x$ when $h(x) = 0$.

By practicing different variations, you'll become more comfortable with function notation and problem-solving strategies. Math is like a muscle – the more you exercise it, the stronger it gets! So, keep practicing, and you'll become a function whiz in no time!

If you guys have any questions or want to explore other math topics, feel free to ask! Keep learning and keep exploring! You've got this!