Math Problem Statement

Consider the function f(x)=(ax+3)/(x-b) Find a and b given that y = f(x) with equations x=-1 and y=2

Solution

To solve for aa and bb in the function f(x)=ax+3xbf(x) = \frac{ax+3}{x-b}, we use the information provided: when x=1x = -1, y=2y = 2.

Step 1: Substitute x=1x = -1 and y=2y = 2 into the function

The function becomes: 2=a(1)+31b.2 = \frac{a(-1) + 3}{-1 - b}.

Simplify: 2=a+31b.2 = \frac{-a + 3}{-1 - b}.

Multiply through by the denominator 1b-1 - b (assuming b1b \neq -1): 2(1b)=a+3.2(-1 - b) = -a + 3.

Simplify: 22b=a+3.-2 - 2b = -a + 3.

Rearrange to express aa in terms of bb: a=5+2b.a = 5 + 2b.

Step 2: Determine additional conditions

The problem does not give enough information for a unique solution. To find both aa and bb, additional conditions (e.g., another point on the curve) are needed.

Would you like to provide more details or constraints? Alternatively, I can outline typical approaches to deduce values in similar problems. Let me know!


Related Questions:

  1. How can the vertical asymptote x=bx = b help determine bb?
  2. What happens to aa and bb if f(x)f(x) must be continuous at x=0x = 0?
  3. How would the solution change if another (x,y)(x, y) pair is provided?
  4. How can this function's symmetry be analyzed to simplify solving for aa and bb?
  5. Can derivatives of f(x)f(x) provide information about aa and bb?

Tip: Always check for the domain restrictions of a rational function; the denominator must not equal zero.

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Math Problem Analysis

Mathematical Concepts

Algebra
Rational Functions
Equations

Formulas

f(x) = (ax + c) / (x - b)

Theorems

Solving equations by substitution
Domain restrictions of rational functions

Suitable Grade Level

Grades 9-11