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Chapter 1: Problem 10

Find the values for each function, if they exist, then simplify. a. \(f(0)\) b. \(f(1)\) c. \(f(3)\) d. \(f(-x)\) e. \(f(a)\) f. \(f(a+h)\) \(f(x)=|x-7|+8\)

Short Answer

Expert verified
a. 15, b. 14, c. 12, d. \(|x+7|+8\), e. \(|a-7|+8\), f. \(|a+h-7|+8\)

Step by step solution

01

Evaluate f(0)

Substitute 0 into the function.\[ f(0) = |0-7| + 8 \]Simplify the absolute value first:\[ f(0) = | -7 | + 8 = 7 + 8 = 15 \]
02

Evaluate f(1)

Substitute 1 into the function.\[ f(1) = |1-7| + 8 \]Simplify the expression inside the absolute value:\[ f(1) = | -6 | + 8 = 6 + 8 = 14 \]
03

Evaluate f(3)

Substitute 3 into the function.\[ f(3) = |3-7| + 8 \]Simplify the expression inside the absolute value:\[ f(3) = | -4 | + 8 = 4 + 8 = 12 \]
04

Evaluate f(-x)

Substitute -x into the function. \[ f(-x) = |-x-7| + 8 \]Simplify the absolute value expression given \(|-x| = |x|\):\[ f(-x) = |-x-7| + 8 = |x+7| + 8 \]
05

Evaluate f(a)

Substitute a into the function. \[ f(a) = |a-7| + 8 \]This is already simplified, as it cannot be further reduced without a specific value for a.
06

Evaluate f(a+h)

Substitute a+h into the function. \[ f(a+h) = |a+h-7| + 8 \]This remains in simplified form unless specific values for a or h are provided.

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Key Concepts

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

Function Evaluation
Function evaluation involves substituting specific values into a function and performing the indicated operations to determine the output. Here, our main function is given as \( f(x) = |x-7| + 8 \). This function involves an absolute value, which measures the distance a number is from zero on a number line, always giving a non-negative result.
Let's break this down further:
  • To evaluate the function, replace \( x \) with the desired value inside the function's formula. For instance, to find \( f(0) \), substitute 0 for \( x \).
  • Using this substitution, calculate the expression inside the absolute value to find its magnitude, ignoring its sign.
  • Add the constant part of the function (in our case, 8) to this absolute value to find the full evaluation.
A clear understanding of substitution and maintaining the correct order of operations provides accurate results when evaluating functions.
Algebraic Expressions
An algebraic expression is a mathematical phrase that includes numbers, variables, and operations. In our function, we see an expression within the absolute value, \(|x-7|\).
Understanding algebraic expressions is crucial for evaluating functions:
  • The expression \( |x-7| \) consists of the variable \( x \) and a constant \( -7 \). The absolute value indicates we are interested in the magnitude of \((x-7)\), treating negative results as positive.
  • A constant component, like +8, is simply arithmetic added to the expression within the absolute value.
  • Substitution affects only the variable within the expression. For example, replacing \( x \) with 3 gives us \(|3-7| + 8 \), making the inside of the absolute \(|-4|\) which simplifies to 4.
  • Keep variable places in mind when substituting symbols, negative signs around variables as with \(f(-x)\) influence how the expression unfolds but absolute values will normalize signs.
Mastery of algebraic expressions and their components empowers our ability to handle substitution and solve function-related problems accurately.
Function Simplification
Function simplification is the process of reducing a function's expression to its simplest form. Often, this involves resolving any absolute values and combining like terms if possible.
For the function \( f(x) = |x-7| + 8 \):
  • The absolute value is simplified by converting negative results to their positive counterparts. For example, \(|1-7| = |-6| = 6\).
  • Once the absolute value is simplified, simply add the remaining constant, 8, to this result which yields the final evaluation, like \(6 + 8 = 14\) for \(f(1)\).
  • In cases without specific values, like \(f(a)\) or \(f(a+h)\), simplification halts since no further reduction is possible without actual numerical inputs for these variables.
The overarching idea of function simplification is about transforming expressions into a form that is as concise and clear as possible. Every step of this process requires a careful application of mathematical rules. This makes it easier to understand and work with functions across various contexts.

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