If f(x) = a^x + b where a, b ∈ ℝ, and f(0) = 3 and f(1) = 5, then what are the values of a and b?

Understand the Problem

The question is asking to determine the values of 'a' and 'b' in the function defined as f(x) = a^x + b, given the conditions that f(0) = 3 and f(1) = 5. This involves substituting the given values into the function to create a system of equations and solving for 'a' and 'b'.

Answer

The values are $ a = 3 $ and $ b = 2 $.

Steps to Solve

Set up the function equations

Given the function ( f(x) = a^x + b ), we can plug in the values of ( x ) based on the conditions provided:

For ( f(0) = 3 ):

$$ f(0) = a^0 + b = 1 + b $$
Therefore, we create the first equation:

$$ 1 + b = 3 $$

Solve for ( b )

Rearranging the first equation to solve for ( b ):

$$ b = 3 - 1 $$

Thus,

$$ b = 2 $$

Set up the second function equation

Next, using the second condition ( f(1) = 5 ):

For ( f(1) = 5 ):

$$ f(1) = a^1 + b = a + b $$
Now substitute ( b = 2 ):

$$ a + 2 = 5 $$

Solve for ( a )

Rearranging this equation, we find ( a ):

$$ a = 5 - 2 $$

Therefore,

$$ a = 3 $$

The values are ( a = 3 ) and ( b = 2 ).

More Information

Thus, the solution to the equations yields ( a ) as 3 and ( b ) as 2, which operates in the given function based on the input conditions.

Tips

A common mistake is to misinterpret the exponent in ( f(x) = a^x + b ), especially when substituting values of ( x ).
Ensure that each step systematically follows from the equations created, particularly when rearranging and solving for ( a ) and ( b ).

AI-generated content may contain errors. Please verify critical information

Thank you for voting!