Problem 53 - Intermediate Algebra

Question

If \(f(x)=-x^{2}-2 x-7\), find \(f(-a), f(-a-2)\), and \(f(a+7)\).

Step-by-Step Solution

Answer

\(f(-a) = -a^2 + 2a - 7\), \(f(-a-2) = -a^2 - 2a - 7\), \(f(a+7) = -a^2 - 16a - 70\).

Step 1: Substitute for f(-a)

We substitute \(-a\) into the function \(f(x) = -x^2 - 2x - 7\): \[f(-a) = -(-a)^2 - 2(-a) - 7.\] Simplifying: \[-a^2 + 2a - 7.\] Thus, \(f(-a) = -a^2 + 2a - 7.\)

Step 2: Substitute for f(-a-2)

We substitute \(-a-2\) in the function \(f(x) = -x^2 - 2x - 7\): \[f(-a-2) = -(-a-2)^2 - 2(-a-2) - 7.\] First, calculate \((-a-2)^2\):\[(-a-2)^2 = a^2 + 4a + 4.\]Then substitute:\[-(a^2 + 4a + 4) + 2a + 4 - 7.\] Simplifying: \[-a^2 - 4a - 4 + 2a + 4 - 7,\] which reduces to:\(-a^2 - 2a - 7\). Thus, \(f(-a-2) = -a^2 - 2a - 7.\)

Step 3: Substitute for f(a+7)

We substitute \(a+7\) into the function \(f(x) = -x^2 - 2x - 7\): \[f(a+7) = -(a+7)^2 - 2(a+7) - 7.\] First, calculate \((a+7)^2\):\[(a+7)^2 = a^2 + 14a + 49.\]Then substitute:\[-(a^2 + 14a + 49) - 2a - 14 - 7.\] Simplifying: \[-a^2 - 14a - 49 - 2a - 14 - 7,\] which reduces to:\(-a^2 - 16a - 70.\) Thus, \(f(a+7) = -a^2 - 16a - 70.\)

Key Concepts

Function Evaluation

Function evaluation is a key concept in mathematics where we determine the output of a function for given inputs. Think of a function like a machine: you provide it an input, it processes this input, and then gives you an output. If we have a function defined, such as: \[f(x) = -x^2 - 2x - 7\] and we want to find the value of the function at a specific input, we perform a function evaluation.

• To evaluate \(f(-a)\), we substitute \(-a\) in place of \(x\) in the function.
• This substitution allows us to find the polynomial expression for \(f(-a)\) by following simple arithmetic operations.

Understanding function evaluation is crucial, as it forms the basis for exploring more complex mathematical problems. By plugging in different values into the function, we gain different outputs, which can help us visualize and understand the behavior of the function.

Algebraic Expressions

Algebraic expressions are mathematical phrases that can include numbers, variables, and operation symbols. Consider the expression \(-x^2 - 2x - 7\). Here, - \(-x^2\) represents the square of \(x\) multiplied by \(-1\), - \(-2x\) is \(-2\) times \(x\), and - \(-7\) is a constant. These expressions can be manipulated through addition, subtraction, multiplication, and division. When dealing with algebraic expressions:

• Identify the terms (parts of the expression separated by plus or minus signs) and simplify where possible.
• For example, in the expression for \(f(a+7)\), we expanded \((a+7)^2\) using the distributive property to get \(a^2 + 14a + 49\).
• Simplifying expressions involves combining like terms and reducing the equation as much as possible.

Grasping the manipulation of algebraic expressions helps in simplifying problems and finding solutions more efficiently.

Substitution Method

The substitution method is a commonly used technique in algebra to solve equations or evaluate functions. This involves replacing a variable with a specific value, which can be a number or another expression. For example, if we have a function \( f(x) = -x^2 - 2x - 7 \) and we want to find \( f(-a) \), we replace every instance of \(x\) in the function with \(-a\). **The Process of Substitution**

• Identify the expression or value to substitute: Here, it's \(-a\), \(-a-2\), and \(a+7\).
• Replace the variable in the function with this expression: For \(f(-a)\), replace \(x\) with \(-a\) to get \(-(-a)^2 - 2(-a) - 7\).
• Simplify the resulting expression by performing algebraic operations: Find the square of \(-a\), simplify terms, and combine them to get the new expression.

Understanding the substitution method helps in applying functions to different situations, making it easier to work through complex arithmetic and algebraic problems. This approach is not limited to quadratic functions but is a versatile tool across various types of equations.