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Math  /  Algebra

QuestionSuppose that f(x)=1+3xf(x)=1+3^{x}. (a) Sketch a graph of ff. Then sketch the graph of f−1(x)f^{-1}(x) on the same coordinate axes. Remember, inverse functions switch their xx and yy values. To do this, it may be easiest to make a table of values for ff, and then reverse those values (switching xx and yy columns) for f−1f^{-1}.

Studdy Solution

STEP 1

1. The function f(x)=1+3x f(x) = 1 + 3^x is defined for all real numbers x x .
2. The inverse function f−1(x) f^{-1}(x) exists and can be graphed by switching x x and y y values from f(x) f(x) .

STEP 2

1. Create a table of values for f(x) f(x) .
2. Sketch the graph of f(x) f(x) using the table of values.
3. Create a table of values for f−1(x) f^{-1}(x) by switching x x and y y values from f(x) f(x) .
4. Sketch the graph of f−1(x) f^{-1}(x) using the table of values on the same coordinate axes as f(x) f(x) .

STEP 3

Create a table of values for f(x)=1+3x f(x) = 1 + 3^x . Choose a range of x x values, such as x=−2,−1,0,1,2 x = -2, -1, 0, 1, 2 .
xf(x)−21+3−2=1+19≈1.11−11+3−1=1+13≈1.3301+30=1+1=211+31=1+3=421+32=1+9=10\begin{array}{c|c} x & f(x) \\ \hline -2 & 1 + 3^{-2} = 1 + \frac{1}{9} \approx 1.11 \\ -1 & 1 + 3^{-1} = 1 + \frac{1}{3} \approx 1.33 \\ 0 & 1 + 3^0 = 1 + 1 = 2 \\ 1 & 1 + 3^1 = 1 + 3 = 4 \\ 2 & 1 + 3^2 = 1 + 9 = 10 \\ \end{array}

STEP 4

Using the table of values, plot the points (−2,1.11),(−1,1.33),(0,2),(1,4),(2,10)(-2, 1.11), (-1, 1.33), (0, 2), (1, 4), (2, 10) on a coordinate plane.
Draw a smooth curve through these points to represent the graph of f(x)=1+3x f(x) = 1 + 3^x .

STEP 5

Create a table of values for f−1(x) f^{-1}(x) by switching the x x and y y values from the table of f(x) f(x) .
xf−1(x)1.11−21.33−12041102\begin{array}{c|c} x & f^{-1}(x) \\ \hline 1.11 & -2 \\ 1.33 & -1 \\ 2 & 0 \\ 4 & 1 \\ 10 & 2 \\ \end{array}

STEP 6

Using the table of values for f−1(x) f^{-1}(x) , plot the points (1.11,−2),(1.33,−1),(2,0),(4,1),(10,2)(1.11, -2), (1.33, -1), (2, 0), (4, 1), (10, 2) on the same coordinate plane.
Draw a smooth curve through these points to represent the graph of f−1(x) f^{-1}(x) .
The graph of f(x)=1+3x f(x) = 1 + 3^x and its inverse f−1(x) f^{-1}(x) are now sketched on the same coordinate axes.

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