Problem 77 - College Algebra

Question

For the following exercises, use the functions \(f(x)=2 x^{2}+1\) and \(g(x)=3 x+5\) to evaluate or find the composite function as indicated. $$f(g(x))$$

Step-by-Step Solution

Answer

The composite function is \(f(g(x)) = 18x^2 + 60x + 51\).

Step 1: Understanding Composite Functions

A composite function is a function formed by applying one function to the results of another. In this case, we need to find \(f(g(x))\), which means we'll substitute \(g(x)\) into \(f(x)\).

Step 2: Substitute \(g(x)\) Into \(f(x)\)

The first step to find \(f(g(x))\) is to replace every occurrence of \(x\) in \(f(x) = 2x^2 + 1\) with \(g(x) = 3x + 5\). This gives us \(f(g(x)) = f(3x + 5)\).

Step 3: Perform the Substitution

Substitute \(x = 3x + 5\) into \(f(x) = 2x^2 + 1\) to get: \[ f(g(x)) = 2(3x + 5)^2 + 1 \]

Step 4: Expand the Expression

First, find \((3x + 5)^2\): \[ (3x + 5)^2 = (3x + 5)(3x + 5) = 9x^2 + 30x + 25 \].

Step 5: Multiply by 2

Now, multiply the expanded result by 2: \[ 2(9x^2 + 30x + 25) = 18x^2 + 60x + 50 \].

Step 6: Add 1 to the Expression

Finally, add the constant 1 to the expression: \[ f(g(x)) = 18x^2 + 60x + 50 + 1 = 18x^2 + 60x + 51 \].

Step 7: Conclusion

Thus, the composite function \(f(g(x))\) is found to be \(18x^2 + 60x + 51\).

Key Concepts

Substitution in Functions

Substitution in functions is key when dealing with composite functions. It involves replacing the variable in one function with another function. In our case, we need to find the function composition \(f(g(x))\). This means we take the expression from \(g(x) = 3x + 5\) and plug it into the function \(f(x) = 2x^2 + 1\).
This operation might seem complex, but it can be simplified into clear steps:

• Identify where substitution needs to occur. For \(f(g(x))\), substitute \(x\) in \(f(x)\) by the entire expression of \(g(x)\).
• Replace \(x\) in \(f(x)\) with \(g(x)\). This transforms \(f(x)\) into \(f(g(x)) = f(3x + 5)\).

Understanding this substitution process is foundational to successfully manipulating and evaluating composite functions.

Expanding Expressions

Expanding expressions involves breaking down and simplifying algebraic expressions. After substitution, the next step in calculating \(f(g(x))\) involves expanding the squaring operation. Our task is to expand \((3x + 5)^2\).
Here's how it works:

• The expression \((3x + 5)^2\) is expanded by multiplying \((3x + 5)\) by itself: \((3x + 5)(3x + 5)\).
• Use the distributive property, also known as the FOIL method for binomials, to achieve this: multiply each term in the first binomial by each term in the second, resulting in \(9x^2 + 30x + 25\).

It's crucial to carefully expand expressions to obtain accurate results.

Evaluating Functions

Evaluating functions means calculating the value of the function for specific inputs. For composite functions like \(f(g(x))\), each step of the process is an evaluation until the final expression is reached. After expansion, our work is not complete because we have more evaluation to do.
Steps to evaluate include:

• Take the expanded form \((9x^2 + 30x + 25)\), and multiply it by 2, reflecting the coefficient in the original \(f(x)\): \(2(9x^2 + 30x + 25)\).
• This results in \(18x^2 + 60x + 50\).
• Finally, add the constant 1: \(18x^2 + 60x + 50 + 1\).

Make sure each step in the evaluation process aligns with the operations in the original functions; this ensures correctness in the results.

Quadratic Expressions

Quadratic expressions are polynomial expressions of degree 2. They commonly appear in the form \(ax^2 + bx + c\). The final result from our function composition, \(f(g(x)) = 18x^2 + 60x + 51\), is a classic quadratic expression.
Quadratic expressions have distinct features and uses:

• The coefficient \(18\) of \(x^2\) indicates how "steep" the parabola will open if graphed.
• The linear term \(60x\) affects the slope and position of the vertex of the parabola.
• The constant \(51\) determines the y-intercept.

Knowing how to recognize and manipulate quadratic expressions is central in algebra, impacting graphing or solving equations analytically or numerically.