Chapter 2: Q. 41 (page 69)

In Problems 39– 46, find the following for each function:
$(a) f (0) (b) f (1) (c) f (- 1) (d) f (- x) (e) - f (x) (f) f (x + 1) (g) f (x) (h) f (x + h)$
localid="1645960232226" $f (x) = \frac{x}{x^{2} + 1}$

Short Answer

Expert verified

The value of

f (0) = 0 f (1) = \frac{1}{2} f (- 1) = \frac{- 1}{2} f (- x) = \frac{- x}{x^{2} + 1} - f (x) = \frac{- x}{x^{2} + 1} f (x + 1) = \frac{x + 1}{x^{2} + 2 x + 2} f (x) = \frac{x}{x^{2} + 1} f (x + h) = \frac{x + h}{x^{2} + 2 x h + h^{2} + 1}

Step by step solution

Step 1. Given Information

We have to find the value for each function:

$(a) f (0) (b) f (1) (c) f (- 1) (d) f (- x) (e) - f (x) (f) f (x + 1) (g) f (x) (h) f (x + h)$

The given equation is $f (x) = \frac{x}{x^{2} + 1}$

Part (a) Step 1. We find the value of f(0)Putting x=0 in the given question.

$f (0) = \frac{0}{{(0)}^{2} + 1} f (0) = 0$

Part (b) Step 1. We find the value of f(1)Putting x=1 in the given question.

$f (1) = \frac{1}{{(1)}^{2} + 1} f (1) = \frac{1}{1 + 1} f (1) = \frac{1}{2}$

Part (c) Step 1. We find the value of f(-1)Putting x=-1 in the given question.

$f (- 1) = \frac{- 1}{{(- 1)}^{2} + 1} f (- 1) = \frac{- 1}{1 + 1} f (- 1) = \frac{- 1}{2}$

Part (d) Step 1. We find the value of f(-x)Putting x=-x in the given question.

$f (- x) = \frac{- x}{{(- x)}^{2} + 1} f (- x) = \frac{- x}{x^{2} + 1}$

Part (e) Step 1. We find the value of -f(x)To find the value we multiply the given equation by -sign

$- f (x) = - (\frac{x}{x^{2} + 1}) - f (x) = \frac{- x}{x^{2} + 1}$

Part (f) Step 1. We find the value of f(x+1)Putting x=x+1 in the given question.

$f (x + 1) = \frac{x + 1}{{(x + 1)}^{2} + 1}$

Using the formula ${(a + b)}^{2} = a^{2} + 2 a b + b^{2}$

$f (x + 1) = \frac{x + 1}{x^{2} + 2 x + 1 + 1} f (x + 1) = \frac{x + 1}{x^{2} + 2 x + 2}$

Part (g) Step 1. We find the value of f(x)

The value of $f (x) = \frac{x}{x^{2} + 1}$

Part (h) Step 1. We find the value of f(x+h)Putting x=x+h in the given question.

$f (x + h) = \frac{x + h}{{(x + h)}^{2} + 1}$

Using the formula ${(a + b)}^{2} = a^{2} + 2 a b + b^{2}$

$f (x + h) = \frac{x + h}{x^{2} + 2 x h + h^{2} + 1}$

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In Problems 39– 46, find the following for each function:
$(a) f (0) (b) f (1) (c) f (- 1) (d) f (- x) (e) - f (x) (f) f (x + 1) (g) f (x) (h) f (x + h)$
localid="1645960232226" $f (x) = \frac{x}{x^{2} + 1}$

Short Answer

Step by step solution

Step 1. Given Information

Part (a) Step 1. We find the value of f(0)Putting x=0 in the given question.

Part (b) Step 1. We find the value of f(1)Putting x=1 in the given question.

Part (c) Step 1. We find the value of f(-1)Putting x=-1 in the given question.

Part (d) Step 1. We find the value of f(-x)Putting x=-x in the given question.

Part (e) Step 1. We find the value of -f(x)To find the value we multiply the given equation by -sign

Part (f) Step 1. We find the value of f(x+1)Putting x=x+1 in the given question.

Part (g) Step 1. We find the value of f(x)

Part (h) Step 1. We find the value of f(x+h)Putting x=x+h in the given question.

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In Problems 39– 46, find the following for each function:(a)f(0)(b)f(1)(c)f(-1)(d)f(-x)(e)-f(x)(f)f(x+1)(g)f(x)(h)f(x+h)localid="1645960232226" f(x)=xx2+1

Short Answer

Step by step solution

Step 1. Given Information

Part (a) Step 1. We find the value of f(0)Putting x=0 in the given question.

Part (b) Step 1. We find the value of f(1)Putting x=1 in the given question.

Part (c) Step 1. We find the value of f(-1)Putting x=-1 in the given question.

Part (d) Step 1. We find the value of f(-x)Putting x=-x in the given question.

Part (e) Step 1. We find the value of -f(x)To find the value we multiply the given equation by -sign

Part (f) Step 1. We find the value of f(x+1)Putting x=x+1 in the given question.

Part (g) Step 1. We find the value of f(x)

Part (h) Step 1. We find the value of f(x+h)Putting x=x+h in the given question.

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Math Textbooks

Statistics

Pure Maths

Theoretical and Mathematical Physics

Calculus

Mechanics Maths

Logic and Functions

Study anywhere. Anytime. Across all devices.

In Problems 39– 46, find the following for each function:
$(a) f (0) (b) f (1) (c) f (- 1) (d) f (- x) (e) - f (x) (f) f (x + 1) (g) f (x) (h) f (x + h)$
localid="1645960232226" $f (x) = \frac{x}{x^{2} + 1}$