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Chapter 0: Problem 13

If \(f(x)=x^{2}-3 x\), find \(f(0), f(5), f(3)\), and \(f(-7)\).

Short Answer

Expert verified
f(0)=0, f(5)=10, f(3)=0, f(-7)=70.

Step by step solution

01

- Calculate f(0)

To find the value of the function at 0, substitute 0 into the function: \( f(0) = 0^2 - 3 \times 0 = 0 - 0 = 0 \)
02

- Calculate f(5)

To find the value of the function at 5, substitute 5 into the function: \( f(5) = 5^2 - 3 \times 5 = 25 - 15 = 10 \)
03

- Calculate f(3)

To find the value of the function at 3, substitute 3 into the function: \( f(3) = 3^2 - 3 \times 3 = 9 - 9 = 0 \)
04

- Calculate f(-7)

To find the value of the function at -7, substitute -7 into the function: \( f(-7) = (-7)^2 - 3 \times (-7) = 49 + 21 = 70 \)

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Key Concepts

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

polynomial functions
A polynomial function is an expression involving variables and constants, combined using addition, subtraction, and multiplication. For instance, in the exercise given, the function is defined as \(f(x) = x^2 - 3x\). Here, the terms involve a variable, \(x\), raised to the power of 2 (which creates a quadratic polynomial), and another term where \(x\) is raised to the power of 1. The constants are multiplied directly to these variables or their squares. Polynomial functions can have many terms, but the one in our example has just two terms.
substitution method
The substitution method involves replacing the variable in the function with a specific number to find the function's value at that point. In our exercise, we're substituting specific values into the polynomial to find the result. For example, to find \(f(0)\), we replace \(x\) with 0 in the function. So, we calculate \( f(0) = 0^2 - 3 \times 0 = 0 - 0 = 0\). This method helps us see how the function behaves at specific points.
step-by-step solutions
Breaking down a problem into smaller steps makes it easier to understand and solve. Here, each function evaluation is given step-by-step:
  • To find \(f(0)\), substitute 0 in place of \(x\) in the expression, giving a straightforward result of zero.
  • For \(f(5)\), replace \(x\) with 5: \(5^2 = 25\) and \(3 \times 5 = 15\), thus \(25 - 15 = 10\).
  • To find \(f(3)\), substitute 3: \(3^2 = 9\) and \(3 \times 3 = 9\), thus \(9 - 9 = 0\).
  • Finally, for \(f(-7)\), replace \(x\) with -7: \((-7)^2 = 49\) and \(-3 \times -7 = 21\), thus \(49 + 21 = 70\).
This systematic approach ensures that none of the steps are skipped, ensuring better understanding and accuracy.
quadratic functions
A quadratic function is a type of polynomial where the highest exponent of the variable is 2. The general form is \(ax^2 + bx + c\). In our example, the quadratic function is given as \(f(x) = x^2 - 3x\). These functions form parabolas when graphed. The vertex and direction (upwards or downwards) depend on the coefficients of \(x^2\), \(x\), and the constant term. Quadratic functions are a common type of polynomial function studied in algebra and are essential for understanding more advanced mathematics.

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