Problem 94 - College Algebra Essentials

Question

let \(f(x)=x^{2}-x+4\) and \(g(x)=3 x-5\) Find \(g(-1)\) and \(f(g(-1))\)

Step-by-Step Solution

Answer

The value of \(g(-1)\) is \(-8\) and \(f(g(-1))\) is \(76\).

Step 1: Calculate \(g(-1)\)

To solve \(g(-1)\), substitute \(-1\) into \(g(x)=3x-5\). Calculation will be like \(g(-1)=3(-1)-5\).

Step 2: Evaluate \(g(-1)\)

After substitution, the calculation becomes \(-3-5\), which evaluates to \(-8\). Therefore, \(g(-1) = -8\).

Step 3: Substitute \(g(-1)\) into \(f(x)\)

To solve \(f(g(-1))\), substitute the result of \(g(-1)\) into \(f(x)=x^2-x+4\). The calculation should be expressed as \(f(g(-1))=f(-8)=(-8)^2-(-8)+4\).

Step 4: Evaluate \(f(g(-1))\)

Evaluate the expression. Initially, \(-8^2\) is equal to 64, and \(-(-8)\) becomes 8. So, \(f(g(-1))\) equals \(64+8+4=76\)

Key Concepts

Polynomial Functions

Polynomial functions are mathematical expressions that consist of variables raised to whole number powers and their coefficients. A polynomial can be as simple as a linear function like a line, or as complex as a high-degree polynomial.

In our exercise, we have the polynomial function defined as \(f(x) = x^2 - x + 4\). This is a quadratic polynomial, which typically forms a parabolic shape when graphed.

Characteristics of polynomial functions include:

• They contain terms that are a sum of variables raised to non-negative integer powers.
• The coefficients are real numbers.
• These can have one or more terms, based on the degree, like linear (degree 1), quadratic (degree 2), cubic (degree 3), etc.

Recognizing polynomial structures aids in function evaluation and further manipulation of expressions.

Function Evaluation

Function evaluation involves finding the output of a function given a specific input. It is like following a set of instructions to get to a result. For instance, if you know your function is \( g(x) = 3x - 5 \), substituting \( x = -1 \) allows you to evaluate it as \( g(-1) = 3(-1) - 5 \).

In our exercise, this yields \(-8\), meaning when \(x=-1\) is plugged into the function \(g(x)\), the output is \(-8\). It’s like inputting a value and cranking out a number via the equation.

Function evaluation is a fundamental skill, necessary for solving more complex problems in algebra and calculus.

• Evaluating a function can help in graphing functions to determine their shape and behavior.
• It is crucial for verifying solutions to equations and inequalities.

It helps bridge understanding from abstract concepts to real values.

Substitution Method

The substitution method is a technique widely used to solve equations and evaluate functions. It involves replacing a variable with a given value or another expression. Think of it as swapping out parts of an equation to simplify or solve it.

In our given problem, substitution helps find \( f(g(-1)) \). First, solve \(g(-1)\) which results in \(-8\). Next, substitute \(-8\) into the function \(f(x) = x^2 - x + 4\). This means you're calculating \(f(-8)\), which equals \(64 + 8 + 4\), giving us \(76\).

Substitution streamlines the process of handling complex composite functions.

• It helps simplify expressions by handling one part at a time.
• This method is indispensable when dealing with systems of equations or compound expressions.

Mastering this technique eases the process of breaking down and solving intricate mathematical problems.