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Solved on Jan 17, 2024

Solve −2x=0.5x-2^{x}=0.5 x by finding the values of xx where the functions f(x)=−2xf(x)=-2^{x} and g(x)=0.5xg(x)=0.5 x intersect.

STEP 1

Assumptions
1. We are given two functions, f(x)=−2xf(x) = -2^x and g(x)=0.5xg(x) = 0.5x.
2. We need to evaluate these functions at x=−2,−1,0,1,2x = -2, -1, 0, 1, 2.
3. The outputs should be expressed as decimals if they are fractional values.

STEP 2

Evaluate the function f(x)=−2xf(x) = -2^x at x=−2x = -2.
f(−2)=−2−2f(-2) = -2^{-2}

STEP 3

Calculate the value of −2−2-2^{-2}.
−2−2=−(122)=−14-2^{-2} = -\left(\frac{1}{2^2}\right) = -\frac{1}{4}

STEP 4

Convert the fractional value to a decimal.
−14=−0.25-\frac{1}{4} = -0.25

STEP 5

Evaluate the function f(x)=−2xf(x) = -2^x at x=−1x = -1.
f(−1)=−2−1f(-1) = -2^{-1}

STEP 6

Calculate the value of −2−1-2^{-1}.
−2−1=−(12)=−0.5-2^{-1} = -\left(\frac{1}{2}\right) = -0.5

STEP 7

Evaluate the function f(x)=−2xf(x) = -2^x at x=0x = 0.
f(0)=−20f(0) = -2^{0}

STEP 8

Calculate the value of −20-2^{0}.
−20=−1-2^{0} = -1

STEP 9

Evaluate the function f(x)=−2xf(x) = -2^x at x=1x = 1.
f(1)=−21f(1) = -2^{1}

STEP 10

Calculate the value of −21-2^{1}.
−21=−2-2^{1} = -2

STEP 11

Evaluate the function f(x)=−2xf(x) = -2^x at x=2x = 2.
f(2)=−22f(2) = -2^{2}

STEP 12

Calculate the value of −22-2^{2}.
−22=−4-2^{2} = -4

STEP 13

Evaluate the function g(x)=0.5xg(x) = 0.5x at x=−2x = -2.
g(−2)=0.5⋅(−2)g(-2) = 0.5 \cdot (-2)

STEP 14

Calculate the value of 0.5⋅(−2)0.5 \cdot (-2).
0.5⋅(−2)=−10.5 \cdot (-2) = -1

STEP 15

Evaluate the function g(x)=0.5xg(x) = 0.5x at x=−1x = -1.
g(−1)=0.5⋅(−1)g(-1) = 0.5 \cdot (-1)

STEP 16

Calculate the value of 0.5⋅(−1)0.5 \cdot (-1).
0.5⋅(−1)=−0.50.5 \cdot (-1) = -0.5

STEP 17

Evaluate the function g(x)=0.5xg(x) = 0.5x at x=0x = 0.
g(0)=0.5⋅0g(0) = 0.5 \cdot 0

STEP 18

Calculate the value of 0.5⋅00.5 \cdot 0.
0.5⋅0=00.5 \cdot 0 = 0

STEP 19

Evaluate the function g(x)=0.5xg(x) = 0.5x at x=1x = 1.
g(1)=0.5⋅1g(1) = 0.5 \cdot 1

STEP 20

Calculate the value of 0.5⋅10.5 \cdot 1.
0.5⋅1=0.50.5 \cdot 1 = 0.5

STEP 21

Evaluate the function g(x)=0.5xg(x) = 0.5x at x=2x = 2.
g(2)=0.5⋅2g(2) = 0.5 \cdot 2

STEP 22

Calculate the value of 0.5⋅20.5 \cdot 2.
0.5⋅2=10.5 \cdot 2 = 1

STEP 23

Now that we have evaluated both functions at the given xx values, we can fill in the table.
For f(x)=−2xf(x) = -2^x: - At x=−2x = -2, f(x)=−0.25f(x) = -0.25 - At x=−1x = -1, f(x)=−0.5f(x) = -0.5 - At x=0x = 0, f(x)=−1f(x) = -1 - At x=1x = 1, f(x)=−2f(x) = -2 - At x=2x = 2, f(x)=−4f(x) = -4
For g(x)=0.5xg(x) = 0.5x: - At x=−2x = -2, g(x)=−1g(x) = -1 - At x=−1x = -1, g(x)=−0.5g(x) = -0.5 - At x=0x = 0, g(x)=0g(x) = 0 - At x=1x = 1, g(x)=0.5g(x) = 0.5 - At x=2x = 2, g(x)=1g(x) = 1
The completed table is:
xf(x)=−2xg(x)=0.5x−2−0.25−1−1−0.5−0.50−101−20.52−41\begin{array}{c|c|c} x & f(x)=-2^{x} & g(x)=0.5x \\ \hline -2 & -0.25 & -1 \\ -1 & -0.5 & -0.5 \\ 0 & -1 & 0 \\ 1 & -2 & 0.5 \\ 2 & -4 & 1 \\ \end{array}

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