Question

multiple choice 4 points use a graphing calculator to solve the equation. round your answer to two decimal places. ( e^x = -x ) (\bigcirc) {-1.05} (\bigcirc) {1.05} (\bigcirc) {-0.57} (\bigcirc) {0.57} clear my selection

Explanation:

Step1: Define the functions

We have the equation \( e^{x}=-x \). We can rewrite this as finding the intersection of the two functions \( y = e^{x} \) and \( y=-x \).

Step2: Analyze the functions

The function \( y = e^{x} \) is an exponential function that is always positive (\( e^{x}>0 \) for all real \( x \)) and increasing. The function \( y=-x \) is a linear function with a slope of - 1 and a y - intercept of 0. We are looking for the value of \( x \) where \( e^{x}=-x \). Let's test the values:

• For \( x=- 1.05 \): \( e^{-1.05}=\frac{1}{e^{1.05}}\approx\frac{1}{2.8576}\approx0.35 \), and \( -x = 1.05 \). Since \( 0.35

eq1.05 \), this is not a solution.

• For \( x = 1.05 \): \( e^{1.05}\approx2.8576 \), and \( -x=-1.05 \). Since \( 2.8576

eq - 1.05 \), this is not a solution.

• For \( x=-0.57 \): \( e^{-0.57}=\frac{1}{e^{0.57}}\approx\frac{1}{1.768}\approx0.565 \), and \( -x = 0.57 \). Rounding \( 0.565 \) to two decimal places gives \( 0.57 \), so \( e^{-0.57}\approx0.57=-(-0.57) \).
• For \( x = 0.57 \): \( e^{0.57}\approx1.768 \), and \( -x=-0.57 \). Since \( 1.768

eq - 0.57 \), this is not a solution.

Answer:

\(\{-0.57\}\)