An algebraic function F(a, b) is defined as: F(a, b) = \frac{a^2 - 3b}{a b^{-1} + 2} The values for variables a and b are given in the table below: | Variable | Value | | :--- | :--- | | a | -4 | | b | -\frac{1}{2} | What is the value of F(a, b) for these given values?

An algebraic function F(a, b) is defined as: F(a, b) = \frac{a^2 - 3b}{a b^{-1} + 2} The v… — ACT

An algebraic function

F(a, b)

is defined as:

F(a, b) = \frac{a^2 - 3b}{a b^{-1} + 2}

The values for variables $a$ and $b$ are given in the table below:

Variable	Value
$a$	$-4$
$b$	$-\frac{1}{2}$

What is the value of $F(a, b)$ for these given values?

The value of

F(a, b)

\frac{7}{4}

Substituting

a = -4

and

b = -1/2

into the expression gives a numerator of

(-4)^2 - 3(-1/2) = 16 + 3/2 = 35/2

and a denominator of

(-4)(-2) + 2 = 8 + 2 = 10

. Dividing

35/2

10

simplifies to

35/20

, which equals

7/4

Evaluate the reciprocal term

b^{-1}

b^{-1} = \left(-\frac{1}{2}\right)^{-1} = -2

A negative exponent indicates taking the reciprocal of the base.

Evaluate the numerator

a^2 - 3b

(-4)^2 - 3\left(-\frac{1}{2}\right) = 16 + \frac{3}{2} = \frac{35}{2}

Squaring a negative number yields a positive value (

(-4)^2 = 16

), and subtracting a negative value is equivalent to addition.

Evaluate the denominator

a b^{-1} + 2

(-4)(-2) + 2 = 8 + 2 = 10

Multiplying two negative numbers yields a positive product.

Divide the numerator by the denominator and simplify

\frac{\frac{35}{2}}{10} = \frac{35}{20} = \frac{7}{4}

Dividing

\frac{35}{2}

10

gives

\frac{35}{20}

, which simplifies to

\frac{7}{4}

when dividing both numerator and denominator by

5

Evaluating algebraic expressions with negative numbers, fractions, and negative exponents.

Estimated Time:1m 30s