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Unformatted text preview: Math 112- A - Answers 1. (10 pts.) Sketch the graph of f ( x ) = 2 x +2- 5 and find the following: (a) Domain: (-∞ , ∞ ) (b) Range: (- 5 , ∞ ), (it is the range of 2 x , (0 , ∞ ) shifted 5 units down). (c) Asymptote: y =- 5 (d) y- intercept EXACT: f (0) = 2 2- 5 =- 1 (e) x- intercept EXACT: Solution to f ( x ) = 0: 2 x +2- 5 = 0, 2 x +2 = 5, x + 2 = log 2 5, x = log 2 5- 2. 2. (6 pts.) Find the exact domain of f ( x ) = ln( x 2- 5). The restriction on the domain of the logarithm is that it only can be applied to positive numbers, so we impose the inequality: x 2- 5 > 0, then solve it: x 2 > 5, so | x | > √ 5. Hence, the domain is (-∞ , √ 5) ∪ ( √ 5 , ∞ ). 3. (12 pts.) Find the inverse of the following functions: Following the method: equate to y , switch x and y , and then solve for y : (a) f ( x ) = 3+4 x 5- 2 x = y , 3+4 y 5- 2 y = x , 3 + 4 y = x (5- 2 y ), 3 + 4 y = 5 x- 2 xy , 4 y + 2 xy = 5 x- 3, (4 + 2 x ) y = 5 x- 3, y = 5 x- 3 2 x +4 . f- 1 ( x ) = 5 x- 3 2 x + 4 . (b) g ( x ) = ln( x- 3) + 2 = y , ln( y- 3) + 2 = x , ln( y- 3) = x- 2, y- 3 = e x- 2 , y = e x- 2 + 3. g- 1 ( x ) = e x- 2 + 3. 4. (10 pts.) Given that log 5 2 = . 43, and that log 5 3 = . 68, fill in the table with the appropriate values: x 6 10 2 / 3 8 / 5 9 log 5 x 1 . 11 1 . 43- . 25 . 29 1 . 36 1 Math 112- A - Answers log 5 6 = log 5 (2 · 3) = log 5 2 + log 5 3 = . 43 + . 68 = 1 . 11 log 5 10 = log 5 (2 ·...
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