Problem 64 - Elementary and Intermediate Algebra

Question

Find each function value, if possible. Do not use a calculator. See Example 5 . \(h(t)=\sqrt{t^{2}+t-3}\) a. \(h(-4)\) b. \(h(-1)\)

Step-by-Step Solution

Answer

h(-4) = 3; h(-1) is not defined.

Step 1: Understand the Function

The function given is \( h(t) = \sqrt{t^2 + t - 3} \). This is a square root function, which implies the expression inside the square root must be non-negative (i.e., \( t^2 + t - 3 \geq 0 \)) for \( h(t) \) to be real and defined.

Step 2: Evaluate h(-4)

Substitute \( t = -4 \) into the function: \[ h(-4) = \sqrt{(-4)^2 + (-4) - 3} \]Compute inside the square root: \[ h(-4) = \sqrt{16 - 4 - 3} = \sqrt{9} \]Since \( \sqrt{9} = 3 \), we have \( h(-4) = 3 \).

Step 3: Evaluate h(-1)

Substitute \( t = -1 \) into the function: \[ h(-1) = \sqrt{(-1)^2 + (-1) - 3} \]Compute inside the square root: \[ h(-1) = \sqrt{1 - 1 - 3} = \sqrt{-3} \]The expression \( \sqrt{-3} \) is not defined in the real number system as it is a square root of a negative number.

Key Concepts

Square Root Function

The square root function is a fundamental mathematical concept used to find the value that, when multiplied by itself, gives the original number. It is denoted as \( \sqrt{x} \) and is only defined for non-negative numbers in the real number system. This means the expression under the square root must be zero or positive. * For example, \( \sqrt{9} = 3 \) because \( 3\times3 = 9 \).* If the value inside the square root is negative, the result is not a real number.In the function \( h(t) = \sqrt{t^2 + t - 3} \), the expression \( t^2 + t - 3 \) must be non-negative to find a real and defined value. Evaluating the function at different values involves substituting \( t \) and checking if the expression remains non-negative. If negative, the value isn't part of the real number system, leading to an undefined result.

Real Number System

The real number system is a set of numbers including both rational and irrational numbers, that can exist on the number line. Real numbers include positive numbers, negative numbers, and zero. However, they do not include imaginary numbers, such as those arising with the square root of a negative number. * Rationale: Includes numbers that can be expressed as a fraction or ratio, such as \( \frac{1}{2} \) or \( 3 \).* Irrational: Numbers that cannot be expressed as simple fractions, like \( \pi \) or \( \sqrt{2} \).In the given function evaluation, we encountered \( \sqrt{-3} \), which does not belong to the real number system as it leads to an imaginary number. This emphasizes the importance of ensuring the expression under a square root remains non-negative when determining the value in a real number context.

Substitution Method

In mathematics, the substitution method involves replacing a variable in an expression or equation with a given value. This allows for straightforward evaluation by simplifying the expression. Here's how you apply it in function evaluation:1. Identify the given variable in the function.2. Replace it with the specified number.3. Simplify the expression to find the result.For our function \( h(t) = \sqrt{t^2 + t - 3} \): * When evaluating \( h(-4) \), substitute \( t \) with \(-4\), resulting in \( \sqrt{9} = 3 \). * Similarly, in evaluating \( h(-1) \), substitute \( t \) with \(-1\), resulting in \( \sqrt{-3} \), illustrating an undefined result in the real number system. The substitution method is crucial in determining the output of functions given specific values, emphasizing understanding algebraic manipulations and numeric results.