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Math  /  Algebra

QuestionSuppose that g(x)=3x+3\mathrm{g}(\mathrm{x})=3^{\mathrm{x}}+3. (a) What is g(−1)\mathrm{g}(-1) ? When x=−1\mathrm{x}=-1, what is the point on the graph of g ? (b) If g(x)=12\mathrm{g}(\mathrm{x})=12, what is x ? When g(x)=12\mathrm{g}(\mathrm{x})=12, what is the point on the graph of g ?

Studdy Solution

STEP 1

1. The function g(x)=3x+3 g(x) = 3^x + 3 is given.
2. We need to evaluate g(−1) g(-1) and find the corresponding point on the graph.
3. We need to solve for x x when g(x)=12 g(x) = 12 and find the corresponding point on the graph.

STEP 2

1. Evaluate g(−1) g(-1) and find the point on the graph.
2. Solve the equation g(x)=12 g(x) = 12 and find the point on the graph.

STEP 3

Evaluate g(−1) g(-1) .
Substitute x=−1 x = -1 into the function g(x) g(x) :
g(−1)=3−1+3 g(-1) = 3^{-1} + 3
Calculate 3−1 3^{-1} :
3−1=13 3^{-1} = \frac{1}{3}
Substitute back into the equation:
g(−1)=13+3 g(-1) = \frac{1}{3} + 3
Convert 3 to a fraction with a denominator of 3:
3=93 3 = \frac{9}{3}
Add the fractions:
g(−1)=13+93=103 g(-1) = \frac{1}{3} + \frac{9}{3} = \frac{10}{3}

STEP 4

Find the point on the graph when x=−1 x = -1 .
The point on the graph is (−1,103)(-1, \frac{10}{3}).

STEP 5

Solve the equation g(x)=12 g(x) = 12 .
Set the function equal to 12:
3x+3=12 3^x + 3 = 12
Subtract 3 from both sides:
3x=12−3 3^x = 12 - 3 3x=9 3^x = 9
Recognize that 9 is a power of 3:
9=32 9 = 3^2
So, we have:
3x=32 3^x = 3^2
Since the bases are the same, set the exponents equal:
x=2 x = 2

STEP 6

Find the point on the graph when g(x)=12 g(x) = 12 .
The point on the graph is (2,12)(2, 12).

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