Problem 86 Suppose $f$ is a function with... [FREE SOLUTION]

91影视

Precalculus A Prelude to Calculus

Sheldon Axler

$Math Studyset 91影视 Explanations$ Math

1 Edition

Chapter 6: Problem 86

Suppose $f$ is a function with period $p$. Explain why $$ f(x-p)=f(x) $$ for every number $x$ such that $x-p$ is in the domain of $f$.

Short Answer

Expert verified

A function $f$ with period $p$ means that for all $x$ in its domain, $f(x+p) = f(x)$. We want to show that $f(x-p) = f(x)$ when $x-p$ is in the domain of $f$. By the periodicity property, since the difference between $x-p$ and $x$ is a multiple of $p$ (i.e., $p$ itself), we have $f(x-p) = f(x)$. This demonstrates how the function values repeat at intervals of $p$.

Step by step solution

Understanding the definition of period

A function $f$ is said to have a period $p$ if for all $x$ in its domain, $f(x+p) = f(x)$. This means that the function values repeat at intervals of length $p$.

Examining the function with a different input

Now let's consider the value of the function at the input $x-p$: $f(x-p)$. Our goal is to show that $f(x-p) = f(x)$ when $x-p$ is in the domain of $f$.

Using the periodicity property

Since $f$ has period $p$, we know that $f(x) = f(y)$ if the difference between $x$ and $y$ is a multiple of $p$. We can apply this property to the input $x-p$. Let's consider the function value at $x - p$. We have: $$ f(x - p) = f(x - p + p) = f(x), $$ because the difference between $x-p$ and $x$ is exactly $p$.

Conclusion

Therefore, for every number $x$ such that $x-p$ is in the domain of $f$, we have $f(x-p) = f(x)$. This statement shows how the function values repeat at intervals of $p$ and is a direct consequence of the definition of a function having a period $p$.

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91影视

Suppose \(f\) is a function with period \(p\). Explain why $$ f(x-p)=f(x) $$ for every number \(x\) such that \(x-p\) is in the domain of \(f\).

Short Answer

Step by step solution

Understanding the definition of period

Examining the function with a different input

Using the periodicity property

Conclusion

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