Problem 64
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
Verified Answer
a. \( h(-4) = 3 \); b. \( h(-1) \) is undefined.
1Step 1: Substitute the Value of t
Substitute the given value of \( t = -4 \) into the function \( h(t) = \sqrt{t^2 + t - 3} \). This gives us: \[ h(-4) = \sqrt{(-4)^2 + (-4) - 3} \]
2Step 2: Simplify the Expression Inside the Square Root
Simplify the expression inside the square root: \[ (-4)^2 = 16, \]\[ 16 + (-4) = 12, \]\[ 12 - 3 = 9. \] Thus, \[ h(-4) = \sqrt{9}. \]
3Step 3: Evaluate the Square Root
Evaluate the square root: \[ \sqrt{9} = 3. \] Thus, the value is \[ h(-4) = 3. \]
4Step 4: Substitute the Value of t for Part b
Substitute the given value of \( t = -1 \) into the function. This gives us: \[ h(-1) = \sqrt{(-1)^2 + (-1) - 3} \]
5Step 5: Simplify the Expression for Part b
Simplify the expression inside the square root for \( t = -1 \): \[ (-1)^2 = 1, \] \[ 1 + (-1) = 0, \] \[ 0 - 3 = -3. \] Thus, \[ h(-1) = \sqrt{-3}. \]
6Step 6: Determine if the Value is Possible
Since the square root of a negative number is not a real number, \( h(-1) \) is undefined in the set of real numbers.
Key Concepts
Understanding Square RootsSimplifying Expressions Inside the Square RootUnderstanding Undefined Values
Understanding Square Roots
Square roots are a way to find a number that, when multiplied by itself, gives the original number. For example, the square root of 9 (\( \sqrt{9} \)) results in 3, because \( 3 \times 3 = 9 \).
However, square roots apply only to non-negative numbers when we are talking about real numbers, not negative ones.
This is because no real number squared will result in a negative number.
This concept becomes crucial in function evaluation.
For instance, if we have a function like \( h(t) = \sqrt{t^2 + t - 3} \), if you find that the expression inside the square root is negative, as we saw in part b of the exercise, \(h(-1)\), then the square root is not defined for real numbers.
However, square roots apply only to non-negative numbers when we are talking about real numbers, not negative ones.
This is because no real number squared will result in a negative number.
This concept becomes crucial in function evaluation.
For instance, if we have a function like \( h(t) = \sqrt{t^2 + t - 3} \), if you find that the expression inside the square root is negative, as we saw in part b of the exercise, \(h(-1)\), then the square root is not defined for real numbers.
Simplifying Expressions Inside the Square Root
Simplifying expressions involves breaking down a complex mathematical expression into simpler components or steps. This is useful in math when dealing with evaluations:
When we plug a value into this expression, simplifying it step-by-step allows us to see what is being calculated clearly.
In step 5 for \( h(-1)\), you see how each calculation is done one at a time:
- It helps in solving the equation quicker.
- Makes the expression easier to understand.
When we plug a value into this expression, simplifying it step-by-step allows us to see what is being calculated clearly.
In step 5 for \( h(-1)\), you see how each calculation is done one at a time:
- \((-1)^2 = 1\)
- 1 added to \((-1)\) gives 0
- 0 minus 3 results in -3
Understanding Undefined Values
Undefined values in mathematics occur when you try to perform operations that do not yield a valid answer.
A common case is taking square roots of negative numbers, which simply do not exist for real numbers.
In the set of real numbers, there is no way to multiply a real number by itself and get a negative result.
This is why in our exercise, \( h(-1) = \sqrt{-3} \), is undefined.
An undefined value signals that there are restrictions on the inputs of a function to ensure valid outputs.
When you come across an expression inside a square root that is negative, the overall expression is deemed undefined in the real number system.
Being aware of possible undefined values helps in handling functions effectively and highlights the importance of verifying valid inputs.
A common case is taking square roots of negative numbers, which simply do not exist for real numbers.
In the set of real numbers, there is no way to multiply a real number by itself and get a negative result.
This is why in our exercise, \( h(-1) = \sqrt{-3} \), is undefined.
An undefined value signals that there are restrictions on the inputs of a function to ensure valid outputs.
When you come across an expression inside a square root that is negative, the overall expression is deemed undefined in the real number system.
Being aware of possible undefined values helps in handling functions effectively and highlights the importance of verifying valid inputs.
Other exercises in this chapter
Problem 63
Rationalize each denominator. $$ \frac{3}{\sqrt[3]{9}} $$
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A shortstop fields a grounder at a point one-third of the way from second base to third base. How far will he have to throw the ball to make an out at first bas
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