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Chapter 10: Problem 84

If \(f(x)=\sqrt{2 x+3}\) and \(g(x)=\sqrt[3]{x-8}\), find each function value. $$ g(0) $$

Short Answer

Expert verified
The value of \(g(0)\) is \(-2\).

Step by step solution

01

Substitute into Function

To find the value of the function \(g\) at \(x = 0\), substitute 0 into the function \(g(x)\). This gives us \(g(0) = \sqrt[3]{0 - 8}\).
02

Simplify the Expression

Next, simplify the expression inside the cube root. We have \(0 - 8 = -8\). This simplifies \(g(0) = \sqrt[3]{-8}\).
03

Evaluate the Cube Root

Finally, find the cube root of \(-8\). The cube root of \(-8\) is \(-2\), since \((-2) \times (-2) \times (-2) = -8\). Thus, \(g(0) = -2\).

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Key Concepts

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

Square Root Function
A square root function is a type of algebraic function that involves the square root of a variable expression. It can be represented as \( f(x) = \sqrt{x} \) or, more generally, \( f(x) = \sqrt{a \cdot x + b} \). Square root functions produce non-negative outputs for non-negative inputs. This is because the square root of any real number is non-negative.
To visualize a square root function, think of points extending rightward from the origin because it typically covers only the first quadrant of the Cartesian plane when \( a > 0 \) and \( b \) ensures the expression under the root is non-negative. Here are some key points:
  • The domain of a basic square root function \( \sqrt{x} \) is \( x \geq 0 \).
  • The range is also \( y \geq 0 \) because square roots never produce negative results.
  • Graphing it results in a curve starting from the point \( (b/2a, 0) \) if the vertex form is used.
Cube Root Function
The cube root function is another type of algebra function similar to the square root but involves finding the cube root of a given expression. The standard form is \( g(x) = \sqrt[3]{x} \) or \( g(x) = \sqrt[3]{a \cdot x + b} \). The cube root function is particularly interesting because:
  • Unlike square roots, cube roots can produce both negative and positive results.
  • The domain of a cube root function is all real numbers \( x \).
  • The range is also all real numbers \( y \) because you can take the cube root of any real number, positive or negative.
In graphing, a cube root function typically looks like an elongated "S" that passes through the origin. It is symmetric around the origin because if \( f(-x) = -f(x) \), then the function is odd.
Function Evaluation
Function evaluation is the process of finding the value of a function for a specific input. This involves substituting the input value, often represented by \( x \), into the function and simplifying the expression. Here's how it works:
  • Take the function, for example, \( f(x) = \sqrt{2x + 3} \).
  • To evaluate the function for a specific \( x \), like \( x = 4 \), substitute \( 4 \) in place of \( x \).
  • This gives \( f(4) = \sqrt{2 \cdot 4 + 3} = \sqrt{11} \).
This technique is fundamental in algebra and helps determine outputs for given inputs, essentially assessing the behavior of functions at specific points.
Substitution Method
The substitution method is a straightforward technique used to evaluate functions, solve equations, or solve systems of equations. When using this method to evaluate functions, it involves:
  • Taking the given function, like \( g(x) = \sqrt[3]{x - 8} \).
  • Replacing the variable \( x \) with the given input value鈥攊n our case, \( 0 \).
Specifically, substituting \( 0 \) into \( g(x) \) gives you \( g(0) = \sqrt[3]{0 - 8} \).
This results in simplifying the expression, \( \sqrt[3]{-8} \), and finally evaluating it to \( -2 \).
The substitution method is essential in algebra, ensuring the correct input transformation into manageable mathematical expressions.

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