The function f has the property f(x+1)=2x+3+2*f(-1). What are f(-1) and f(x)?

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giorgiana1976 | College Teacher | (Level 3) Valedictorian

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Since the expression of f(x+1) is a linear function (f(-1) is a constant), then f(x) is a linear function.

We'll write the standard form of a linear function:

f(x) = ax + b

We'll calculate f(-1):

f(-1) = -a + b

Now, we'll calculate f(x+1) = a*(x+1) + b

f(x+1) = ax + a + b (1)

But the property of f(x) is f(x+1) = 2x+3+2*f(-1). (2)

Equating (1) and (2), yields:

ax + a + b = 2x + 3 + 2*f(-1)

Comparing, we'll get:

a = 2

a + b = 3 + 2*(-a + b)

We'll replace a by it's value and we'll remove the brackets:

2 + b = 3 - 4 + 2b

We'll shift b to the right side and -1 to the left:

2 + 1 = 2b - b

b = 3

The requested function f(x) is f(x) = 2x + 3 and f(-1) = 1.

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