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Chapter 0: Problem 89

Evaluate \(f(4)\). $$f(x)=x^{2}$$

Short Answer

Expert verified
The value of \(f(4)\) is 16.

Step by step solution

01

Understand the Function

The function provided is a quadratic function given by \(f(x) = x^2\).
02

Substitute the Given Value

To evaluate \(f(4)\), substitute \(x = 4\) into the function. So, \(f(4) = 4^2\).
03

Perform the Calculation

Calculate \(4^2\). This means you need to multiply 4 by itself: \(4 \times 4 = 16\).
04

Write the Final Answer

Thus, \(f(4) = 16\).

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Key Concepts

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

quadratic function
A quadratic function is a type of polynomial function. It has the general form: \[ f(x) = ax^2 + bx + c \] where \(a\), \(b\), and \(c\) are constants, and \(x\) represents the variable. The term \(ax^2\) is what makes it a quadratic function, distinguishing it from linear functions. In the given exercise, the quadratic function is simpler: \[ f(x) = x^2 \] Here, \(a = 1\), and both \(b\) and \(c\) are zero, so it merely squares the input value. Quadratic functions graph as parabolas, which can open upwards or downwards depending on the sign of \(a\). For \(f(x) = x^2\), the parabola opens upwards.
substitution
Substitution is a fundamental technique in algebra where we replace a variable with a given value or another expression. This simplifies the expression for easier calculation. In the example, we have the function \(f(x) = x^2\) and are asked to evaluate it at \(x = 4\).
To do this, we'll substitute \(4\) for \(x\) in the function: \[ f(4) = 4^2 \] This means wherever there is an \(x\) in the function, it gets replaced by \(4\). Substitution helps to transform a general expression into a specific value that can be more easily evaluated. This method is widely used in solving equations and simplifying functions.
basic arithmetic
Basic arithmetic involves simple operations like addition, subtraction, multiplication, and division. For our current problem, we focus on multiplication because we need to calculate \[ 4^2 \] This requires multiplying 4 by itself: \[ 4 \times 4 = 16 \] Doing these calculations accurately is crucial because even a small mistake can lead to the wrong answer. For students, mastery of basic arithmetic sets the foundation for more advanced mathematical concepts.
In this problem, correctly performing the multiplication confirms that: \[ f(4) = 16 \] Thus, getting the right result.

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