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MP DTE Even/Odd Functions — practice questions

Explore 53 free Even/Odd Functions questions in MP DTE Mathematics. Open each question or exercise to read its content, reference answers and available explanation.

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Even/Odd Functions questions and exercises with answers

Showing questions 1–30 of 53.

If $f(x)$ is an **odd** function, then $\int_{-a}^{a} f(x)\,dx$ equals:Read the answer and worked solution →If $f(x)$ is an **even** function, then $\int_{-a}^{a} f(x)\,dx$ equals:Read the answer and worked solution →$\int_{-\pi/2}^{\pi/2} \sin^3 x\,dx$ equals:Read the answer and worked solution →If $f(x) = x^3 \sin^4 x$, then $\int_{-\pi/4}^{\pi/4} f(x)\,dx$ equals:Read the answer and worked solution →$\int_{-2}^{2} (x^3 + x^2)\,dx$ equals:Read the answer and worked solution →Sort each inverse function into its domainRead the answer and worked solution →Tap every function whose domain is [−1, 1]Read the answer and worked solution →Match each inverse function to its domainRead the answer and worked solution →Match each inverse function to its principal value branchRead the answer and worked solution →Flip and pair: function with its principal rangeRead the answer and worked solution →Which principal ranges sit around zero, and which run 0 to π?Read the answer and worked solution →Tap every function whose principal range has a point removedRead the answer and worked solution →Name the principal value branchRead the answer and worked solution →Domain checkRead the answer and worked solution →Put the reasoning in order: why an inverse needs a restricted domainRead the answer and worked solution →True or false: existence of inversesRead the answer and worked solution →True or false: reading the tableRead the answer and worked solution →Spot the one that does not belongRead the answer and worked solution →Two are true, one is notRead the answer and worked solution →Assertion and reasonRead the answer and worked solution →Complete the summary tableRead the answer and worked solution →Complete the rule about restricting a domainRead the answer and worked solution →Three of these are wrong. Find the one that holds.Read the answer and worked solution →Principal values worked in the chapterRead the answer and worked solution →Order the steps for finding a principal valueRead the answer and worked solution →True or false: principal valuesRead the answer and worked solution →Match each identity to the condition it needsRead the answer and worked solution →Assertion and reason: identitiesRead the answer and worked solution →Match each mathematician to the year your syllabus givesRead the answer and worked solution →Trace how trigonometry travelledRead the answer and worked solution →

Worked practice questions

If $f(x)$ is an **odd** function, then $\int_{-a}^{a} f(x)\,dx$ equals:

  1. $2\int_0^a f(x)\,dx$
  2. $0$
  3. $\int_0^a f(x)\,dx$
  4. $-2\int_0^a f(x)\,dx$
Show answer and solution

Correct answer: B. $0$

For odd $f$: $f(-x) = -f(x)$, so contributions from $[-a, 0]$ and $[0, a]$ cancel exactly. **Hence $\int_{-a}^{a} f = 0$.**

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If $f(x)$ is an **even** function, then $\int_{-a}^{a} f(x)\,dx$ equals:

  1. $0$
  2. $\int_0^a f(x)\,dx$
  3. $2\int_0^a f(x)\,dx$
  4. $-2\int_0^a f(x)\,dx$
Show answer and solution

Correct answer: C. $2\int_0^a f(x)\,dx$

For even $f$: $f(-x) = f(x)$, so the contributions from $[-a, 0]$ and $[0, a]$ are equal. **Thus $\int_{-a}^{a} f = 2\int_0^a f$.**

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$\int_{-\pi/2}^{\pi/2} \sin^3 x\,dx$ equals:

  1. $0$
  2. $2/3$
  3. $\pi$
  4. $1$
Show answer and solution

Correct answer: A. $0$

$\sin^3(-x) = -\sin^3 x$, so $\sin^3 x$ is an **odd** function. By Property VI, the integral over the symmetric interval $[-\pi/2, \pi/2]$ is $0$.

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