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Chapter 8: Problem 10

If \(f(x)=x^{2}-7 x\), find \(f(a), f(a-3)\), and \(f(a+h)\).

Short Answer

Expert verified
\(f(a) = a^2 - 7a\), \(f(a-3) = a^2 - 13a + 30\), and \(f(a+h) = a^2 + 2ah + h^2 - 7a - 7h\).

Step by step solution

01

Understanding the Function

The function given is \(f(x) = x^2 - 7x\). This is a quadratic function with \(x\) being the variable.
02

Substituting to Find \(f(a)\)

To find \(f(a)\), substitute \(a\) for \(x\) in \(f(x)\): \[f(a) = a^2 - 7a.\] Thus, the expression for \(f(a)\) is \(a^2 - 7a\).
03

Substituting to Find \(f(a-3)\)

To find \(f(a-3)\), substitute \(a-3\) for \(x\) in \(f(x)\): \[f(a-3) = (a-3)^2 - 7(a-3).\] Simplify this expression: 1. Expand \((a-3)^2\) to get \(a^2 - 6a + 9\).2. Simplify \(-7(a-3)\) to \(-7a + 21\).3. Combine these results: \(a^2 - 6a + 9 - 7a + 21 = a^2 - 13a + 30\).Thus, \(f(a-3) = a^2 - 13a + 30\).
04

Substituting to Find \(f(a+h)\)

To find \(f(a+h)\), substitute \(a+h\) for \(x\) in \(f(x)\): \[f(a+h) = (a+h)^2 - 7(a+h).\] Simplify this expression:1. Expand \((a+h)^2\) to get \(a^2 + 2ah + h^2\).2. Simplify \(-7(a+h)\) to \(-7a - 7h\).3. Combine these results: \(a^2 + 2ah + h^2 - 7a - 7h\).Thus, \(f(a+h) = a^2 + 2ah + h^2 - 7a - 7h\).

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Key Concepts

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

Function Substitution
Function substitution is a method used to evaluate a function at specific values. It's especially useful in exploring relationships or patterns within functions. To achieve function substitution, you replace the variable in the function with the given value or expression.
In the context of our exercise, to find the value of the quadratic function \(f(x) = x^2 - 7x\) when \(x\) is replaced by another expression, we follow substitution steps:
  • Identify the base function: in this case, \(f(x) = x^2 - 7x\).
  • For \(f(a)\), substitute \(a\) for \(x\), giving \(f(a) = a^2 - 7a\).
  • For \(f(a-3)\), substitute \(a-3\) for \(x\): \(f(a-3) = (a-3)^2 - 7(a-3)\).
  • For \(f(a+h)\), substitute \(a+h\) for \(x\): \(f(a+h) = (a+h)^2 - 7(a+h)\).
By using substitution, you manipulate the function's structure to fit the new context or value, providing further insight into its behavior at those points.
Expansion of Binomials
The expansion of binomials is a key algebraic technique that involves multiplying out expressions of the form \((a+b)^2\) or \((a-b)^2\). The process involves applying distributive properties to eliminate parentheses and simplify the expressions.
In this exercise, expansion is needed when computing \(f(a-3)\) and \(f(a+h)\), as they involve expressions like \((a-3)^2\) and \((a+h)^2\):
  • For \((a-3)^2\), distribute to get: \(a^2 - 6a + 9\).
  • For \((a+h)^2\), distribute to get: \(a^2 + 2ah + h^2\).
Each expansion provides a rewritten form of the binomial, which can then be combined with other terms in the expression. Understanding this concept is crucial as it forms the foundation for solving many broader algebraic problems.
Algebraic Expressions
Algebraic expressions involve operations with variables and constants, resulting in expressions that can be simplified to explore their properties. These expressions combine constants, coefficients, and variables adhering to algebraic principles like associative, distributive, and commutative laws.
In our problem, we're dealing with algebraic manipulation while finding \(f(a-3)\) and \(f(a+h)\).
  • Combine and simplify the expressions: After expansion, such as \((a-3)^2\) or \((a+h)^2\), combine similar terms to simplify the quadratic expressions.
  • For \((a-3)^2 - 7(a-3)\), combine terms to get \(a^2 - 13a + 30\).
  • For \((a+h)^2 - 7(a+h)\), combine terms to get \(a^2 + 2ah + h^2 - 7a - 7h\).
Handling algebraic expressions with ease helps in simplifying complicated polynomial expressions, making it a critical skill in solving mathematical problems.

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Most popular questions from this chapter

Explain why the composition of two functions is not a commutative operation.

If \(f(x)=x^{2}\) and \(g(x)=\sqrt{x}\), with both having a domain of the set of nonnegative real numbers, then show that \((f \circ g)(x)=x\) and \((g \circ f)(x)=x\).

Suppose that the viewing window on your graphing calculator is set so that \(-15 \leq x \leq 15\) and \(-10 \leq y \leq 10\). Now try to graph the function \(f(x)=x^{2}-8 x+28\). Nothing appears on the screen, so the parabola must be outside the viewing window. We could arbitrarily expand the window until the parabola appeared. However, let's be a little more systematic and use \(\left(-\frac{b}{2 a}, f\left(-\frac{b}{2 a}\right)\right)\) to find the vertex. We find the vertex is at \((4,12)\), so let's change the \(y\) values of the window so that \(0 \leq y \leq 25\). Now we get a good picture of the parabola. Graph each of the following parabolas, and keep in mind that you may need to change the dimensions of the viewing window to obtain a good picture. (a) \(f(x)=x^{2}-2 x+12\) (b) \(f(x)=-x^{2}-4 x-16\) (c) \(f(x)=x^{2}+12 x+44\) (d) \(f(x)=x^{2}-30 x+229\) (e) \(f(x)=-2 x^{2}+8 x-19\)

This problem is designed to reinforce ideas presented in this section. For each part, first predict the shapes and locations of the parabolas, and then use your graphing calculator to graph them on the same set of axes. (a) \(f(x)=x^{2}, f(x)=x^{2}-4, f(x)=x^{2}+1\), \(f(x)=x^{2}+5\) (b) \(f(x)=x^{2}, f(x)=(x-5)^{2}, f(x)=(x+5)^{2}\), \(f(x)=(x-3)^{2}\) (c) \(f(x)=x^{2}, f(x)=5 x^{2}, f(x)=\frac{1}{3} x^{2}, f(x)=-2 x^{2}\) (d) \(f(x)=x^{2}, f(x)=(x-7)^{2}-3, f(x)=-(x+8)^{2}+\) \(4, f(x)=-3 x^{2}-4\) (e) \(f(x)=x^{2}-4 x-2, f(x)=-x^{2}+4 x+2\), \(f(x)=-x^{2}-16 x-58, f(x)=x^{2}+16 x+58\)

The volume of a cylinder varies jointly as its altitude and the square of the radius of its base. If the volume of a cylinder is 1386 cubic centimeters when the radius of the base is 7 centimeters, and its altitude is 9 centimeters, find the volume of a cylinder that has a base of radius 14 centimeters if the altitude of the cylinder is 5 centimeters.
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