/*! This file is auto-generated */ .wp-block-button__link{color:#fff;background-color:#32373c;border-radius:9999px;box-shadow:none;text-decoration:none;padding:calc(.667em + 2px) calc(1.333em + 2px);font-size:1.125em}.wp-block-file__button{background:#32373c;color:#fff;text-decoration:none} Problem 63 Find \(f(x)\) at the indicated v... [FREE SOLUTION] | 91影视

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Chapter 1: Problem 63

Find \(f(x)\) at the indicated value of \(x\). $$f(x)=\sqrt[3]{x^{2}}, x=8$$

Short Answer

Expert verified
The value of \(f(8)\) is 4.

Step by step solution

01

Substitute the value of x into the function

Substitute the given value of \(x = 8\) into the function \(f(x) = \sqrt[3]{x^2}\). This gives us:\[ f(8) = \sqrt[3]{8^2} \]
02

Simplify the expression

First, calculate \(8^2\) which gives us 64. Substitute this back into the expression:\[ f(8) = \sqrt[3]{64} \]
03

Evaluate the cube root

Now, find the cube root of 64, which is 4, as \(4^3 = 64\). Thus, we have:\[ f(8) = 4 \]

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Cube Roots
Exploring cube roots is an intriguing mathematical journey, as it involves understanding how numbers relate when used as multiplicands three times. Essentially, the cube root of a number is a value that, when multiplied by itself twice, gives you the original number. In mathematical terms, if \( n^3 = x \), then \( n \) is the cube root of \( x \). For example, the cube root of 64 is 4 because \( 4 \times 4 \times 4 = 64 \).
  • Cube roots can be found using different methods like prime factorization or estimation.
  • Understanding cube roots helps in solving higher degree polynomials and algebraic expressions.
In computational terms, the cube root is frequently expressed using the notation \( \sqrt[3]{x} \), and this signifies raising \( x \) to the power of \( \frac{1}{3} \). Recognizing this notation is key for evaluating functions involving cube roots.
Substitution in Functions
Substitution in functions is a practical tool in mathematics. It involves replacing variables in a given function with specific values to determine the result. This process is crucial for evaluating functions at particular points. Let's simplify this with a quick example. Suppose you have a function \( f(x) = x + 3 \). If you need to evaluate \( f(2) \), you replace \( x \) with 2, resulting in \( f(2) = 2 + 3 = 5 \).
  • This technique simplifies the evaluation process by focusing on one variable at a time.
  • It is extensively used in calculus and algebra for function evaluations at specific points.
By consistently practicing substitution, you can develop a deeper understanding and quick reflexes when dealing with more complex functions.
Simplifying Expressions
Simplifying expressions is a fundamental skill that forms the backbone of many mathematical solutions. It involves reducing complex expressions into their simplest form without changing their value. This process can make subsequent calculations much easier and more efficient.
  • To simplify an expression, you often combine like terms or use mathematical properties such as distribution.
  • For instance, in the expression \( 8^2 \), simplifying by calculating gives \( 64 \).
When you stay organized and follow rules for operations鈥攍ike handling exponents and roots appropriately鈥攕implifying expressions becomes more intuitive. It becomes easier over time as you familiarize yourself with algebraic rules and patterns, enabling you to tackle problems of increasing complexity.

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Most popular questions from this chapter

Solve each equation analytically. Check it analytically, and then support the solution graphically. $$\frac{5}{6} x-2 x+\frac{1}{3}=\frac{1}{3}$$

Solve each inequality analytically, writing the solution set in interval notation. Support your answer graphically. (Hint: Once part (a) is done, the answer to part (b) follows.) (a) \(6+3(1-x) \geq 0\) (b) \(6+3(1-x)<0\)

Solve each inequality analytically. Write the solution set in interval notation. Support the answer graphically. $$\frac{1}{3} x-\frac{1}{5} x \leq 2$$

Solve each inequality analytically. Write the solution set in interval notation. Support the answer graphically. $$0.6 x-2(0.5 x+0.2) \leq 0.4-0.3 x$$

Function \(f\) gives the median 2015 weekly income (in dollars) by educational attainment for people 25 years old and over. This function is defined by \(f(N)=493, f(H)=678, f(B)=1137\) and \(f(M)=1341,\) where \(N\) denotes no high school diploma, \(H\) a high school diploma, \(B\) a bachelor's degree, and \(M\) a master's degree. (Source: U.S. Bureau of Labor Statistics.) (a) Write \(f\) as a set of ordered pairs. (b) Give the domain and range of \(f\). (c) Discuss the relationship between education and income.
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