The function f is defined by f x = a x + b , where a and b are constants

Question

The function f is defined by f x = a x + b , where a and b are constants and a &#62; 0 . In the xy -plane, the graph of y = f x has a y -intercept at 0 , - 25 and passes through the point 2 , 23 . What is the value of a + b ?

Accepted Answer

The correct answer is - 19 . It's given that function f is defined by f x = a x + b , where a and b are constants and a > 0 . It's also given that the graph of y = f x in the xy -plane has a y -intercept at 0 , - 25 and passes through the point 2 , 23 . Since the graph has a y -intercept at 0 , - 25 , f 0 = - 25 . Substituting 0 for x in the given equation yields f 0 = a 0 + b , or f 0 = 1 + b , and substituting - 25 for f 0 in this equation yields - 25 = 1 + b . Subtracting 1 from each side of this equation yields - 26 = b . Substituting - 26 for b in the equation f x = a x + b yields f x = a x - 26 . Since the graph also passes through the point 2 , 23 , f 2 = 23 . Substituting 2 for x in the equation f x = a x - 26 yields f 2 = a 2 - 26 , and substituting 23 for f 2 yields 23 = a 2 - 26 . Adding 26 to each side of this equation yields 49 = a 2 . Taking the square root of both sides of this equation yields &plusmn; 7 = a . Since it's given that a > 0 , the value of a is 7 . It follows that the value of a + b is 7 - 26 , or - 19 .

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