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Chapter 7: Problem 90

Find the specified function value, if it exists. $$ g(x)=-\sqrt[3]{2 x-1} ; g(0), g(-62), g(-13), g(63) $$

Short Answer

Expert verified
g(0) = 1, g(-62) = 5, g(-13) = 3, g(63) = -5

Step by step solution

01

Understand the given function

The function given is \[ g(x) = -\sqrt[3]{2x - 1} \] which involves a cube root. This function is defined for all real numbers.
02

Substitute x = 0

To find \( g(0) \), substitute \( x = 0 \) into the function: \[ g(0) = -\sqrt[3]{2(0) - 1} = -\sqrt[3]{-1} = -(-1) = 1 \]
03

Substitute x = -62

To find \( g(-62) \), substitute \( x = -62 \) into the function: \[ g(-62) = -\sqrt[3]{2(-62) - 1} = -\sqrt[3]{-125} = -(-5) = 5 \]
04

Substitute x = -13

To find \( g(-13) \), substitute \( x = -13 \) into the function: \[ g(-13) = -\sqrt[3]{2(-13) - 1} = -\sqrt[3]{-27} = -(-3) = 3 \]
05

Substitute x = 63

To find \( g(63) \), substitute \( x = 63 \) into the function: \[ g(63) = -\sqrt[3]{2(63) - 1} = -\sqrt[3]{125} = -5 \]

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Key Concepts

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

cube root function
A cube root function is one of the fundamental types of functions students must understand in algebra and calculus. It can be written in the form \( f(x) = \sqrt[3]{x} \), where the cube root of \( x \) is taken. Unlike square roots, cube roots are defined for all real numbers鈥攂oth positive and negative. This is because cubing a negative number results in a negative number. For example, the cube root of 27 is 3, since \( 3^3 = 27 \). Similarly, the cube root of -8 is -2, because \( -2^3 = -8 \). This function transforms the values in specific ways, which is essential in understanding more complex functions, including those involving cube roots.
-function evaluation
Function evaluation is the process of determining the value of a function at a specific input. For the given function, \( g(x) = -\sqrt[3]{2x - 1} \), we substitute different values for \( x \):
  • For \( g(0) \), we substitute \(0\) into the function: \( g(0) = -\sqrt[3]{2 \cdot 0 - 1} = -\sqrt[3]{-1} = 1 \). This is because \( -1 \) is the cube root of \( -1 \) and taking the negative outside the function flips the sign.
  • For \( g(-62) \), we substitute \(-62 \) into the function: \( g(-62) = -\sqrt[3]{2 \cdot -62 - 1} = -\sqrt[3]{-125} = 5 \).
  • Similarly, we can substitute other values like \( g(-13) \) and \( g(63) \) by following the same steps.
Function evaluation is crucial for understanding how functions behave at different points and determining their outputs. This method is particularly useful for graphing and solving equations.
substitution method
The substitution method is a straightforward technique used in algebra to simplify function evaluation. Using this method involves replacing the variable in the function with a specific value.
Here are the steps to apply the substitution method:
  • Identify the given function. For example, \( g(x) = -\sqrt[3]{2x - 1} \).
  • Choose the value of \( x \) you want to evaluate. For instance, \( x = 0 \).
  • Substitute the chosen value into the function in place of \( x \). So, for \( g(0) \), replace every occurrence of \( x \) with \( 0 \).
  • Calculate the resulting expression to get the function's value at that specific \( x \). So, \( g(0) = -\sqrt[3]{2 \cdot 0 - 1} = -\sqrt[3]{-1} = 1 \)
Using the substitution method can help simplify complex problems and aid in understanding how functions work. By substituting specific values, you can see how the function's output changes, deepening your comprehension of the function's behavior.

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