Problem 61 - A Graphical Approach to Precalculus with Limits

Question

Find \(f(x)\) at the indicated value of \(x\). $$f(x)=\sqrt{x^{3}+12} ; x=-2$$

Step-by-Step Solution

Answer

The function value is 2 when \(x = -2\).

Step 1: Substitute the value of x

First, we need to replace the value of \(x\) in the function \(f(x)=\sqrt{x^3+12}\) with the value \(x=-2\). This gives us \(f(-2)=\sqrt{(-2)^3+12}\).

Step 2: Calculate the exponent

Next, we calculate \((-2)^3\). Raising \(-2\) to the power of 3 gives \(-8\) because \((-2)\times(-2)\times(-2) = -8\).

Step 3: Simplify inside the square root

Substitute \(-8\) back into the function in place of \((-2)^3\), giving us \(\sqrt{-8+12}\). Simplify inside the square root: \(-8 + 12 = 4\).

Step 4: Simplify the square root

Now calculate \(\sqrt{4}\). The square root of \(4\) is \(2\), because \(2\times2=4\).

Step 5: Conclusion

The function value at \(x=-2\) is \(f(-2)=2\).

Key Concepts

Understanding Square Roots

Square roots are quite intuitive once you get the hang of them. A square root attempts to find a number which, when multiplied by itself, results in the given value.
In mathematical terms, the square root of a number "a" is written as \( \sqrt{a} \). If \( b \) is the square root of \( a \), it should satisfy \( b \times b = a \).
For example, the square root of 4 is 2, since \( 2 \times 2 = 4 \). Similarly, the square root of 9 would be 3, because \( 3 \times 3 = 9 \).

• Finding square roots involves determining which number squared gives you the number you started with.
• For perfect squares like 4, 9, and 16, finding the square root is straightforward.
• If the number isn’t a perfect square, it may result in an irrational number.

In the function \( f(x)=\sqrt{x^3+12} \), once the calculations inside the square root yielded 4, we could simply "take" the square root and find that it equals 2.

The Power of Exponents

Exponents are another essential concept in math. They denote repeated multiplication. An exponent tells you how many times to multiply a number by itself.
The expression \( x^n \) signifies that the number \( x \) is multiplied by itself \( n \) times. For example, \( 3^2 = 3 \times 3 = 9 \). These concepts are the cornerstone for operations in algebra and functions.

• Exponents simplify the notation of multiplying the same number several times.
• Negative exponents flip the number to its reciprocal form.
• Zero as an exponent will always result in the number 1, regardless of the base number.

In the equation \( f(x)=\sqrt{x^3+12} \), when we evaluate \((-2)^3\), we're calculating \((-2)\times(-2)\times(-2)\) and the result is \(-8\). Understanding that this simplifies to \( -8 \) is crucial for further solving.

Function Evaluation Simplified

Function evaluation is an important process in mathematics, where you compute the value of a function for a given input. The function \( f(x) \) represents a relationship where each input \( x \) is mapped to a specific output \( f(x) \).
To evaluate a function, you substitute the given value of \( x \) into the function's expression. This substitution allows us to numerically determine what the function yields for a particular \( x \).

• Function notation like \( f(x) \) is used to denote the operation that will be conducted on the variable \( x \).
• Substitute the target value into the function to examine the outcome.
• Evaluate step-by-step operations within the function to obtain the final result.

In this exercise, substituting \( x = -2 \) into the function \( f(x) = \sqrt{x^3+12} \) provided a path to evaluate \( f(-2) \). This led us to calculate each step into arriving at the output value 2, ensuring we analyzed each part of the expression thoroughly and correctly.