Question

If \(f(x)=2 x+8\) and \(g(x)=3 x-2\), determine each of the following. a) \(f(g(1))\) b) \(f(g(-2))\) \(g(f(-4))\) d) \(g(f(1))\)

Short answer

a) 10, b) -8, c) -2, d) 28

Step-by-step solution

Evaluate g(1)

Start with the inner function for part (a). Calculate the value of the function g at x = 1. \[ g(1) = 3(1) - 2 = 3 - 2 = 1 \] So, \( g(1) = 1 \).

Evaluate f(g(1))

Now use the result from Step 1 as the input for the function f. Since \( g(1) = 1 \), find \( f(1) \). \[ f(1) = 2(1) + 8 = 2 + 8 = 10 \] So, \( f(g(1)) = 10 \).

Evaluate g(-2)

For part (b), start by evaluating the inner function g at x = -2. \[ g(-2) = 3(-2) - 2 = -6 - 2 = -8 \] So, \( g(-2) = -8 \).

Evaluate f(g(-2))

Next, use the result from Step 3 as the input for the function f. Since \( g(-2) = -8 \), find \( f(-8) \). \[ f(-8) = 2(-8) + 8 = -16 + 8 = -8 \] So, \( f(g(-2)) = -8 \).

Evaluate f(-4)

For part (c), start by evaluating the inner function f at x = -4. \[ f(-4) = 2(-4) + 8 = -8 + 8 = 0 \] So, \( f(-4) = 0 \).

Evaluate g(f(-4))

Next, use the result from Step 5 as the input for the function g. Since \( f(-4) = 0 \), find \( g(0) \). \[ g(0) = 3(0) - 2 = 0 - 2 = -2 \] So, \( g(f(-4)) = -2 \).

Evaluate f(1)

For part (d), start by evaluating the inner function f at x = 1. \[ f(1) = 2(1) + 8 = 2 + 8 = 10 \] So, \( f(1) = 10 \).

Evaluate g(f(1))

Next, use the result from Step 7 as the input for the function g. Since \( f(1) = 10 \), find \( g(10) \). \[ g(10) = 3(10) - 2 = 30 - 2 = 28 \] So, \( g(f(1)) = 28 \).

Key concepts

Function Evaluation

Function evaluation is the process of determining the output of a function for a given input. This is the first step before diving into more complex concepts like composite or nested functions. Consider a function like \ f(x) = 2x + 8 \ and an input value. To find the output of the function for a specific x-value, substitute the input into the function’s equation. For instance, to evaluate f(2):
\[ f(2) = 2(2) + 8 = 4 + 8 = 12 \]
This process can help solve more intricate function relationships and is pivotal in precalculus problems.

Nested Functions

Nested functions involve evaluating one function within another function. They require multiple steps and careful tracking of values. Take the example of finding \( f(g(1)) \). First, evaluate g at x = 1:

• \[ g(1) = 3(1) - 2 = 1 \]

Then, use this result as the input for f:

• \[ f(1) = 2(1) + 8 = 10 \]

It’s essential to methodically handle each function separately before combining them. The consistency in following steps prevents errors and ensures correct evaluations.

Composite Functions

Composite functions combine two functions where one function's output becomes another function's input. They’re denoted as \( (f \circ g)(x) \), meaning f(g(x)). To solve, start with the inner function and proceed outward. Here’s an example problem: Find \( (f \circ g)(-2) \):
Step 1: Evaluate the inner function g at x = -2:

• \[ g(-2) = 3(-2) - 2 = -8 \]

Step 2: Use this output as the input to f:

• \[ f(-8) = 2(-8) + 8 = -8 \]

This consistent approach helps manage and solve for composite functions in broader precalculus contexts.

Precalculus Problems

Precalculus problems often integrate multiple algebraic concepts, combining simple function evaluations and composite functions. They develop mathematical intuition and prepare students for calculus. Consider solving \( g(f(-4)) \):
Step 1: Evaluate f at x = -4:

• \[ f(-4) = 2(-4) + 8 = 0 \]

Step 2: Use this result within g:

• \[ g(0) = 3(0) - 2 = -2 \]

Tackling these comprehensive problems enhances algebraic skills and builds a foundation for more advanced studies in mathematics.