Questions: Find the following for the function f(x) = x/(x^2+1) (a) f(0) (b) f(7) (c) f(-7) (d) f(-x) (e) -f(x) (f) f(x+5) (g) f(3x) (h) f(x+h)

Transcript text: Find the following for the function f(x) = $\frac{x}{x^2+1}$ (a) f(0) (b) f(7) (c) f(-7) (d) f(-x) (e) -f(x) (f) f(x+5) (g) f(3x) (h) f(x+h)

Solution

Solution Steps

Step 1: Calculate $ f(0) $

Substitute $ x = 0 $ into the function $ f(x) = \frac{x}{x^2 + 1} $: \[ f(0) = \frac{0}{0^2 + 1} = \frac{0}{1} = 0 \]

Step 2: Calculate $ f(7) $

Substitute $ x = 7 $ into the function: \[ f(7) = \frac{7}{7^2 + 1} = \frac{7}{49 + 1} = \frac{7}{50} \]

Step 3: Calculate $ f(-7) $

Substitute $ x = -7 $ into the function: \[ f(-7) = \frac{-7}{(-7)^2 + 1} = \frac{-7}{49 + 1} = \frac{-7}{50} \]

Final Answer

(a) $ \boxed{0} $
(b) $ \boxed{\frac{7}{50}} $
(c) $ \boxed{-\frac{7}{50}} $
(d) $ \boxed{\frac{-x}{x^2 + 1}} $
(e) $ \boxed{-\frac{x}{x^2 + 1}} $
(f) $ \boxed{\frac{x + 5}{(x + 5)^2 + 1}} $
(g) $ \boxed{\frac{3x}{(3x)^2 + 1}} $
(h) $ \boxed{\frac{x + h}{(x + h)^2 + 1}} $

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