Set 7: Exponential Functions (Intermediate)

Choice A is correct. Choice A is the correct answer. Use the algebraic identity. 1. Let: $a = e^x$ and $b = e^{-x}$, so $a + b = 5$. 2. Note: $ab = e^x \cdot e^{-x} = e^0 = 1$. 3. Identity: $(a+b)^2 - (a-b)^2 = 4 ab$. 4. Substitute: $25- (a-b)^2 = 4(1) = 4$. 5. Solve: $(a-b)^2 = 21$, so $a - b = \pm\sqrt{21}$. 6. Wait: Alternatively: $(e^x + e^{-x})^2 = e^{2 x} + 2 + e^{-2 x} = 25$ $e^{2 x} + e^{-2 x} = 23$ $(e^x - e^{-x})^2 = e^{2 x} - 2 + e^{-2 x} = 23 - 2 = 21$ $e^x - e^{-x} = \pm\sqrt{21}$ So the answer is A, not D! Strategic Tip: $(a \pm b)^2 = a^2 \pm 2 ab + b^2$.