Question

$$ f(x)=\frac{10-x^2}{x^2} $$ What is the value of \(f(x)\) when \(x\) is \(-3\) ? (A) 9 (B) 10 (C) \(\frac{1}{9}\) (D) \(\frac{19}{9}\)

Short answer

(C) \(\frac{1}{9}\)

Step-by-step solution

Write down the function

The function is: \(f(x)=\frac{10-x^2}{x^2}\)

Replace x with -3

Substitute the value \(x=-3\) into the function: \(f(-3)=\frac{10-(-3)^2}{(-3)^2}\)

Simplify the function

Calculate the value of the function with the given value of x: \(f(-3)=\frac{10-(-3)^2}{(-3)^2}\) \(f(-3)=\frac{10-9}{9}\) \(f(-3)=\frac{1}{9}\) So, the value of \(f(-3)\) is \(\frac{1}{9}\). The correct answer is (C) \(\frac{1}{9}\).

Key concepts

Quadratic Functions

A quadratic function is a type of polynomial that is characterized by the term with the highest degree being a square ( x^2). These functions typically follow the form \( ax^2 + bx + c \), where \( a \), \( b \), and \( c \) are constants, and \( a eq 0 \). Quadratics are widely used in mathematics to model situations where relationships between quantities can be expressed as parabolas.

• The standard form is \( ax^2 + bx + c = 0 \).
• The graph of a quadratic function is a parabola which can open upwards or downwards depending on the sign of \( a \).

In our original exercise, the quadratic expression is the numerator of the rational function \( 10 - x^2 \). This is a simple difference of squares form which can be expanded using \((a+b)(a-b)\) if needed, but here, it is simply part of a larger function where \( x^2 \) forms both the numerator and the denominator in the rational expression.

Substitution

Substitution is a principle often used in algebra to replace a variable with a specific value. This process helps to evaluate the expression or equation at a particular point. In our original exercise, substitution allows us to compute the value of the function \( f(x) = \frac{10-x^2}{x^2} \) at \( x = -3 \).

• First, identify the variable to replace, such as \( x \) in the function.
• Insert the given number, in this case \( -3 \), wherever you see \( x \) in the function.

Substituting \( x = -3 \) into the function yields \( f(-3) = \frac{10 - (-3)^2}{(-3)^2} \). This simplifies the equation greatly, helping you to then evaluate this rational expression.

Simplification

Simplification involves making a math expression easier to understand or solve by reducing it to its simplest form. For rational expressions, simplification can be particularly useful to make evaluations manageable.

• Perform arithmetic operations like addition, subtraction, or division within expressions.
• Combine like terms or evaluate expressions with specific numbers.

In the original solution, simplifying \( 10 - (-3)^2 \) gives \( 10 - 9 \) which results in \( 1 \). Further, calculating \( (-3)^2 \) results in \( 9 \), which places the simplification of the fraction as \( \frac{1}{9} \). By simplifying step by step, we correctly solved for \( f(-3) \) and concluded that the correct evaluation yields \( \frac{1}{9} \).

Rational Expressions

Rational expressions consist of a numerator and a denominator which are both polynomials. Solving problems involving these expressions often includes understanding both polynomials and fractions.

• A rational function is defined as \( \frac{P(x)}{Q(x)} \) where \( Q(x) eq 0 \).
• It's crucial to identify any values that would make the denominator zero, which are excluded from the domain.

In our example, \( f(x) = \frac{10-x^2}{x^2} \) represents a rational expression. Both the numerator and the denominator have \( x^2 \) terms; however, it's essential to ensure \( x eq 0 \) to keep the function defined. By correctly handling these polynomial expressions, you can substitute values accurately, like replacing \( x \) with \(-3\) to evaluate \( f(x) \). This understanding helps you manipulate and solve rational expressions confidently.