Write an exponential function f represented by the table.
Understand the Problem
The question is asking to derive an exponential function based on the values presented in the table. It involves identifying the form of the function using the given x and y values.
Answer
The exponential function is \( f(x) = 2^{1 - x} \).
Answer for screen readers
The exponential function is given by: $$ f(x) = 2^{1 - x} $$
Steps to Solve
- Identify the general form of the exponential function
An exponential function has the form ( f(x) = a \cdot b^x ), where ( a ) is a constant and ( b ) is the base of the exponential.
- Determine the value of ( a )
From the table, when ( x = 0 ), ( y = 2 ). We can use this to find ( a ): $$ a = f(0) = 2 $$
- Find the value of ( b )
Next, we will use another pair of values from the table. Taking ( x = 1 ) and ( y = 1 ): $$ f(1) = a \cdot b^1 $$ $$ 1 = 2 \cdot b $$ To solve for ( b ): $$ b = \frac{1}{2} $$
- Construct the exponential function
Now we can write the exponential function using the values of ( a ) and ( b ): $$ f(x) = 2 \left( \frac{1}{2} \right)^x $$
- Simplify the function
We can express it as: $$ f(x) = 2^{1 - x} $$
The exponential function is given by: $$ f(x) = 2^{1 - x} $$
More Information
This function represents a decreasing exponential function, showing how ( y ) halves with each increase in ( x ). The initial value ( f(0) = 2 ) indicates that when ( x = 0 ), the function starts at 2.
Tips
- Mixing up the values of ( a ) and ( b ). Always carefully check which corresponds to the initial value and the base of the exponent.
- Forgetting that ( b ) must be positive and different from 1 for an exponential function.
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