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Chapter 2: Problem 72

Let \(f(x)=3 x^{2}-x\) and \(g(x)=4 x-2 .\) Find the following. $$f(x+h)$$

Short Answer

Expert verified
f(x+h) = 3x^2 + 6xh + 3h^2 - x - h

Step by step solution

01

- Understand Given Functions

We are given the functions \[ f(x) = 3x^2 - x \] and \[ g(x) = 4x - 2 \] and need to find \( f(x+h) \).
02

- Substitute \( x+h \) into \( f(x) \)

To find \( f(x+h) \), substitute \( x+h \) for \( x \) in the function \( f(x) \): \[ f(x + h) = 3(x + h)^2 - (x + h) \]
03

- Expand the Squared Term

Expand \( (x + h)^2 \) using the binomial theorem: \[ (x + h)^2 = x^2 + 2xh + h^2 \]Then substitute this back into the function: \[ f(x + h) = 3(x^2 + 2xh + h^2) - (x + h) \]
04

- Distribute and Simplify

Distribute the 3 and simplify: \[ f(x + h) = 3x^2 + 6xh + 3h^2 - x - h \]

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Key Concepts

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

algebraic substitution
Algebraic substitution is a very important tool in mathematics. It allows us to replace a variable with an expression or another variable. In this problem, we used substitution to find the value of the function at a different point.
Here's how it works:
  • First, understand the original function, which is given by \(f(x) = 3x^2 - x\).
  • Next, we need to find \(f(x+h)\). To do this, we substitute \(x+h\) for the variable \(x\) in our original function. This means wherever we see \(x\) in the function, we replace it with \(x+h\).
  • After substitution, we have the expression \(f(x+h) = 3(x+h)^2 - (x+h)\).
By mastering algebraic substitution, we can simplify and transform complex expressions efficiently.
binomial theorem
The binomial theorem helps us expand expressions that are raised to a power. In this exercise, it is used to expand \((x+h)^2\).
According to the binomial theorem, \((x + h)^2\) can be expanded as:
\[ (x + h)^2 = x^2 + 2xh + h^2 \]
  • This allows us to break down \((x+h)^2\) into simpler terms that we can work with.

Using the binomial theorem is crucial in this step, as it leads us to the expanded form that we can substitute back into the function.
expanding expressions
After using the binomial theorem to expand \((x+h)^2\), we continue shaping the function by expanding the entire expression.
Starting from:
\[ f(x+h) = 3(x^2 + 2xh + h^2) - (x + h) \]
  • Distribute the 3 across each term inside the parenthesis:

    \[ = 3x^2 + 6xh + 3h^2 - (x + h) \]
This process is called expanding. We take a complex looking expression and break it down into simpler parts.
simplifying expressions
Simplifying expressions is the last step where we combine like terms and simplify the expression to make it easier to understand.
  • In this problem, after expanding the expression, we combine like terms:

    \[ 3x^2 + 6xh + 3h^2 - x - h \]
  • We don't have any more like terms here, so this is our simplified expression:
    \[ f(x+h) = 3x^2 + 6xh + 3h^2 - x - h \]
Simplifying makes complex equations easier to work with and understand.

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Most popular questions from this chapter

Solve each problem. The Phillips curve shows the relationship between the unemployment rate \(x\) and the inflation rate \(y .\) If the equation of the curve is \(y=1-\sqrt{x}\) for a certain Third World country, then for what values of \(x\) is the inflation rate less than \(50 \% ?\) (GRAPH CAN'T COPY)

Determine algebraically whether the function is even, odd, or neither. Discuss the symmetry of each function. $$f(x)=1+\frac{1}{x^{2}}$$

Make a table listing ordered pairs for each function. Then sketch the graph and state the domain and range. Identify any intervals on which \(f\) is increasing, decreasing, or constant. $$f(x)=-\sqrt{1-x^{2}}$$

Determine algebraically whether the function is even, odd, or neither. Discuss the symmetry of each function. $$f(x)=\sqrt{x}$$

Write the equation of each graph after the indicated transformation\((s)\) The graph of \(y=\sqrt{x}\) is stretched by a factor of \(3,\) translated five units upward, then reflected in the \(x\) -axis.
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