Problem 78 - College Algebra

Question

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

Step-by-Step Solution

Answer

The value of \( g(f(-3)) \) is 62.

Step 1: Understanding the Given Functions

We are given two functions: \( f(x) = 2x^2 + 1 \) and \( g(x) = 3x + 5 \). We need to find \( g(f(-3)) \), which requires us to first find \( f(-3) \) and then substitute this result into \( g(x) \).

Step 2: Finding \( f(-3) \)

To find \( f(-3) \), we substitute \( -3 \) for \( x \) into the function \( f(x) = 2x^2 + 1 \).\[f(-3) = 2(-3)^2 + 1 = 2(9) + 1 = 18 + 1 = 19\]

Step 3: Substituting into \( g(x) \)

Now that we have \( f(-3) = 19 \), we substitute this value into the function \( g(x) = 3x + 5 \) by replacing \( x \) with 19.\[g(19) = 3(19) + 5 = 57 + 5 = 62\]

Step 4: Final Answer

Thus, the composite function \( g(f(-3)) \) evaluates to 62.

Key Concepts

Function Evaluation

Function Evaluation is the process of determining the output of a function for a particular input. To evaluate a function, you substitute the given input value into the function's equation and simplify it to find the output.
You might think of functions like machines: you feed it a value, and it spits out a result. It's a straightforward yet crucial concept in mathematics. Let's consider the functions we're dealing with:

• If we have the function \( f(x) = 2x^2 + 1 \), to evaluate \( f(-3) \), we plug in -3 for \( x \).
• Similarly, with \( g(x) = 3x + 5 \), if we want \( g(f(-3)) \), first evaluate \( f(-3) \) and use that result as the input for \( g(x) \).

Function evaluation is a fundamental skill that builds the foundation for more complex operations like finding composite functions.

Quadratic Functions

Quadratic Functions are polynomial functions of degree 2. They have the general format \( f(x) = ax^2 + bx + c \), where \( a \), \( b \), and \( c \) are constants and \( a eq 0 \). These functions create a U-shaped curve called a parabola when graphed.In our exercise, we're dealing with \( f(x) = 2x^2 + 1 \), a simple quadratic function.

• The term \( 2x^2 \) is the quadratic term, which dictates the curvature and direction of the parabola. The coefficient \( 2 \) makes it narrow.
• There's no linear term (\( bx \), where \( b = 0 \)), simplifying our equation.
• The constant term \( 1 \) moves the parabola up by 1 unit on the y-axis.

When evaluating quadratic functions like \( f(-3) \), remember to square the input, multiply by \( a \), and add \( c \). Conquering quadratic functions is crucial for effectively handling diverse mathematical problems.

Linear Functions

Linear Functions form straight lines when graphed. They are the simplest type of polynomial functions and have the general formula \( g(x) = mx + c \), where \( m \) is the slope and \( c \) is the y-intercept. For our exercise, we are using the linear function, \( g(x) = 3x + 5 \).

• Here, \( 3 \) is the slope, indicating that for every unit increase in \( x \), \( g(x) \) increases by 3 units.
• The constant \( 5 \) shifts the entire line up by 5 units on the graph.

Understanding linear functions is important because they provide a foundational step on which more complex function types are built. In our exercise when we substitute \( x \) with 19 in \( g(x) \), we are scaling our input by the slope and then adjusting by the intercept, providing clarity on manipulation within composite functions.