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Chapter 11: Problem 75

Find each function value. \(g(x)=x^{2}, g(2)\)

Short Answer

Expert verified
The function value is 4.

Step by step solution

01

Understand the Problem

We need to find the value of the function \( g(x) = x^2 \) when \( x = 2 \).
02

Substitute the Given Value

Substitute the value \( x = 2 \) into the function \( g(x) = x^2 \). This gives us the expression \( g(2) = 2^2 \).
03

Simplify the Expression

Calculate \( 2^2 \) which is \( 2 \times 2 = 4 \).
04

Write the Result

Therefore, the value of the function \( g(x) \) when \( x = 2 \) is \( g(2) = 4 \).

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Key Concepts

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

Substitution
In mathematics, substitution is a crucial process used to simplify or solve equations and expressions. It involves replacing a variable with a specific value. In the context of function evaluation, substitution helps us find the function's output for a particular input.

In our example, we start by replacing the variable \(x\) in the function \(g(x) = x^2\) with the number \(2\). This gives us the new function:
  • \(g(2) = 2^2\)
This step is straightforward but is key to transforming the problem from one involving variables to one involving numbers, which is easier to calculate. Substitution helps us see the direct effect that the input has on the function's result.
Quadratic Function
A quadratic function is a type of polynomial function with a degree of two. It takes the general form \( ax^2 + bx + c \), where \(a\), \(b\), and \(c\) are constants and \(a eq 0\).

In our exercise, the function given is \(g(x) = x^2\), which is one of the simplest forms of a quadratic function. It simplifies the process of evaluating the function because there are no linear or constant terms to consider.
  • The term \(x^2\) signifies that for given inputs, the output is proportional to the square of the input.
  • This quadratic function forms a parabolic curve, opening upwards, when graphed on the Cartesian plane.
Understanding the shape and nature of quadratic functions helps in visualizing and predicting how changes in \(x\) affect \(g(x)\).
Simplification
Simplification is the process of reducing an expression or equation to its simplest form. It is an essential skill for solving mathematical problems efficiently. By simplifying expressions, you can make them easier to understand and solve.

In the provided exercise, simplification comes into play after substitution, where the expression \(2^2\) is calculated.
  • Breaking it down, \(2^2\) means multiplying \(2\) by itself: \(2 \times 2\).
  • The result, \(4\), is the simplest form of this expression.
Simplifying expressions ensures that the final result is clear and concise. This step concludes the evaluation by providing a clear numerical output for \(g(x)\) at \(x = 2\). This practice is not only useful but necessary for reliably obtaining solutions in many areas of mathematics.

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Most popular questions from this chapter

Write a quadratic equation with the given roots. Write the equation in the form \(a x^{2}+b x+c=0,\) where \(a, b,\) and \(c\) are integers. \(6,-6\)

Expand each power. $$ (t+2)^{6} $$

Expand each power. $$ (a-2)^{4} $$

Prove that each statement is true for all positive integers. $$ 1+5+9+\cdots+(4 n-3)=n(2 n-1) $$

State whether each statement is true or false when \(n=1\). Explain. \(7^{n}-3^{n}\) is divisible by 4
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