Problem 27
Question
In \(23-34,\) evaluate each function for the given value. Be sure to show your work. $$ f(x)=\left(\frac{1}{x}+\frac{3}{2}\right)^{-2} ; f(2) $$
Step-by-Step Solution
Verified Answer
The function evaluated at \(x=2\) is \(f(2) = \frac{1}{4}\).
1Step 1: Substitute the Given Value into the Function
The function is given as \( f(x)=\left(\frac{1}{x}+\frac{3}{2}\right)^{-2} \). We need to evaluate this function at \( x=2 \). Begin by substituting \( x = 2 \) into the function. We have: \[ f(2) = \left(\frac{1}{2} + \frac{3}{2}\right)^{-2} \].
2Step 2: Simplify the Expression Inside the Parentheses
Simplify the expression inside the parentheses: \( \frac{1}{2} + \frac{3}{2} \). Since both fractions have the same denominator, simply add the numerators: \( \frac{1+3}{2} = \frac{4}{2} \). This simplifies to \( 2 \). Substitute this back into the function expression: \[ f(2) = (2)^{-2} \].
3Step 3: Apply the Negative Exponent
Recall that a negative exponent means taking the reciprocal and then raising it to the positive exponent. For \( 2^{-2} \), first find the reciprocal of 2, which is \( \frac{1}{2} \), and then square it: \( \left(\frac{1}{2}\right)^2 \).
4Step 4: Calculate the Square
Calculate \( \left(\frac{1}{2}\right)^2 \): \[ \left(\frac{1}{2}\right)^2 = \frac{1^2}{2^2} = \frac{1}{4} \]. Therefore, \( f(2) = \frac{1}{4} \).
Key Concepts
Fraction SimplificationNegative ExponentReciprocalSubstitution in Functions
Fraction Simplification
Fraction simplification is an important skill when dealing with fractions in mathematical expressions. It involves reducing a fraction to its simplest form by dividing the numerator and the denominator by their greatest common divisor. This ensures calculations are easier and more accurate.
When working with an expression like \( \frac{1}{2} + \frac{3}{2} \), both fractions have the same denominator. You can add the numerators directly: \( 1 + 3 = 4 \), keeping the common denominator: \( \frac{4}{2} \), which simplifies to the whole number \( 2 \).
When working with an expression like \( \frac{1}{2} + \frac{3}{2} \), both fractions have the same denominator. You can add the numerators directly: \( 1 + 3 = 4 \), keeping the common denominator: \( \frac{4}{2} \), which simplifies to the whole number \( 2 \).
- Ensure the fractions being added or subtracted have a common denominator.
- Simplify the final fraction by dividing the numerator and denominator by their greatest common divisor.
Negative Exponent
Dealing with negative exponents might seem tricky, but it's straightforward when you know the rule: convert it into a positive exponent by taking the reciprocal of the base. For instance, when you encounter \( (2)^{-2} \), recognize that the negative exponent indicates a reciprocal.
The first step is to find the reciprocal of \( 2 \), which is \( \frac{1}{2} \). You then raise the reciprocal to the positive of the given exponent, squaring it: \( (\frac{1}{2})^2 \).
The first step is to find the reciprocal of \( 2 \), which is \( \frac{1}{2} \). You then raise the reciprocal to the positive of the given exponent, squaring it: \( (\frac{1}{2})^2 \).
- Identify that a negative exponent suggests a reciprocal.
- Convert the negative exponent to a positive by switching to the reciprocal.
- Proceed with the calculation using the positive exponent.
Reciprocal
A reciprocal is a simple flip of any given non-zero number. It is fundamental for handling expressions with negative exponents, like in this problem. To find the reciprocal of a fraction, you simply swap the positions of the numerator and denominator. For example, the reciprocal of \( 2 \) is \( \frac{1}{2} \).
Using reciprocals allows for easier handling of inverse relationships in mathematics, particularly when dealing with exponents and division.
Using reciprocals allows for easier handling of inverse relationships in mathematics, particularly when dealing with exponents and division.
- To find a reciprocal, switch the numerator and the denominator.
- The product of a number and its reciprocal is always \( 1 \).
- Reciprocals are crucial for simplifying expressions involving negative exponents.
Substitution in Functions
Function evaluation often requires substitution, which involves replacing the variable in a function with a given value. For example, when evaluating \( f(x) = \left(\frac{1}{x} + \frac{3}{2}\right)^{-2} \) at \( x = 2 \), you substitute \( 2 \) for \( x \) to get:
\[ f(2) = \left(\frac{1}{2} + \frac{3}{2}\right)^{-2} \]
This substitution transforms the function into an expression you can simplify step by step.
\[ f(2) = \left(\frac{1}{2} + \frac{3}{2}\right)^{-2} \]
This substitution transforms the function into an expression you can simplify step by step.
- Start by inserting the given value in place of the variable.
- Simplify the resulting expression following standard arithmetic rules.
- Continue transforming and simplifying until you achieve the final result.
Other exercises in this chapter
Problem 26
If the area of one face of a cube is \(B\) and the volume of the cube is \(V,\) express \(B\) in terms of \(V\)
View solution Problem 26
Simplify each expression. In each exercise, all variables are positive. \(\frac{\left(a^{x}\right)^{y} b}{a^{x y}}\)
View solution Problem 27
In \(3-37,\) express each power as a rational number in simplest form. $$ 32 \times 32^{\frac{1}{3}} $$
View solution Problem 27
Solve each equation and check. \(100^{x}=1,000^{x-1}\)
View solution