EVEN AND ODD FUNCTIONS WORKSHEET

Problem 1-12 : Determine if f(x) is even or odd function.

Problem 1 :

f(x) = 2x³

Problem 2 :

f(x) = 3x² - 5

Problem 3 :

f(x) = x³ + 5x 

Problem 4 :

f(x) = 2x³ + 3x² - 1

Problem 5 :

f(x) = x⁴ - 2x² + 7

Problem 6 :

f(x) = sinx

Problem 7 :

f(x) = cscx

Problem 8 :

f(x) = secx

Problem 9 :

f(x) = cosx

Problem 10 :

f(x) = tanx

Problem 11 :

f(x) = sinx + tanx

Problem 12 :

f(x) = secx + cosx

Problem 13 :

f(x) = sinx - cscx

Problem 14 :

f(x) = sinx - cscx

Problem 15 : 

Let the point (5, -2) is on the graph of y = f(x).

Name another point on the graph if

(a) f(x) is an even function

(b) f(x) is an odd function

Answers

1. Answer :

f(x) = 2x³

f(-x) = 2(-x)³

f(-x) = 2(-x³)

f(-x) = -2x³

f(-x) = -f(x)

f(x) is an odd function

2. Answer :

f(x) = 3x² - 5

f(-x) = 3(-x)² - 5

f(-x) = 3x² - 5

f(-x) = f(x)

f(x) is an even function

3. Answer :

f(x) = x³ + 5x

f(-x) = (-x)³ + 5(-x)

f(-x) = -x³ - 5x

f(-x) = -(x³ + 5x)

f(-x) = -f(x)

f(x) is an odd function

4. Answer :

f(x) = 2x³ + 3x² - 1

f(-x) = 2(-x)³ + 3(-x)² - 1

f(-x) = 2(-x³) + 3x² - 1

f(-x) = -2x³ + 3x² - 1

f(-x) ≠ f(x) or -f(x)

f(x) is neither even or odd

5. Answer :

f(x) = x⁴ - 2x² + 7

f(-x) = (-x)⁴ - 2(-x)² + 7

f(-x) = x⁴ - 2x² + 7

f(-x) = f(x)

f(x) is an even function

6. Answer :

f(x) = sinx

f(-x) = sin(-x)

Here, the angle is negative (-x). So, it falls in the IV^th quadrant. Since sine is negative in the IV^th quadrant,

f(-x) = -sinx

f(-x) = -f(x)

f(x) is an odd function

7. Answer :

f(x) = cscx

f(-x) = csc(-x)

Since cosecant is negative in the IV^th quadrant,

f(-x) = -cscx

f(-x) = -f(x)

f(x) is an odd function

8. Answer :

f(x) = secx

f(-x) = sec(-x)

Since secant is positive in the IV^th quadrant,

f(-x) = secx

f(-x) = f(x)

f(x) is an even function

9. Answer :

f(x) = cosx

f(-x) = cos(-x)

Since cosine is positive in the IV^th quadrant,

f(-x) = cosx

f(-x) = f(x)

f(x) is an even function

10. Answer :

f(x) = tanx

f(-x) = tan(-x)

Since tangent is negative in the IV^th quadrant,

f(-x) = -tanx

f(-x) = f(x)

f(x) is an odd function

11. Answer :

f(x) = sinx + tanx

f(-x) = sin(-x) + tan(-x)

f(-x) = -sinx - tanx

f(-x) = -(sinx + tanx)

f(-x) = -f(x)

f(x) is an odd function

12. Answer :

f(x) = secx + cosx

f(-x) = sec(-x) + cos(-x)

f(-x) = secx + cosx

f(-x) = f(x)

f(x) is an even function

13. Answer :

f(x) = sinx - cscx

f(-x) = sin(-x) - csc(-x)

f(-x) = -sinx - (-cscx)

f(-x) = -sinx + cscx

f(-x) = -(sinx - cscx)

f(-x) = -f(x)

f(x) is an odd function

14. Answer :

f(x) = cosx - secx

f(-x) = cos(-x) - sec(-x)

f(-x) = cosx - secx

f(-x) = f(x)

f(x) is an even function

14. Answer :

f(x) = cosx - secx

f(-x) = cos(-x) - sec(-x)

f(-x) = cosx - secx

f(-x) = f(x)

f(x) is an even function

15. Answer :

Given : The point (5, -2) is on the graph of y = f(x),

x = 5, y = -2

When x = 5, the value of f(x) is -2.

f(5) = -2 ----(1)

Part (a) :

Given : f(x) is an even function.

f(-x) = f(x)

f(-5) = f(5)

f(-5) = -2 ........from (1)

Another point on the graph is (-5, -2).

Part (b) :

Given : f(x) is an odd function.

f(-x) = -f(x)

f(-5) = -f(5)

f(-5) = -(-2) ........from (1)

f(-5) = 2

Another point on the graph is (-5, 2).

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