Tentukan nilai berikut, jika f(x) = x
3 + 1 dan g(x) = 1
x
untuk
(a) (f + g)(x)
(b) (f − g)(2x)
(c) (f ◦ g)(x)
(d) (g ◦ f)(x)
(e) ((f − g) ◦ g)(x)

Given the functions \( f(x) = x^3 + 1 \) and \( g(x) = \frac{1}{x} \), let's calculate the values for each of the specified operations:

### (a) \((f + g)(x)\)

\[
(f + g)(x) = f(x) + g(x)
\]

\[
= (x^3 + 1) + \frac{1}{x}
\]

\[
= x^3 + 1 + \frac{1}{x}
\]

### (b) \((f - g)(2x)\)

First, find \( f(2x) \) and \( g(2x) \):

\[
f(2x) = (2x)^3 + 1 = 8x^3 + 1
\]

\[
g(2x) = \frac{1}{2x}
\]

Now, calculate \((f - g)(2x)\):

\[
(f - g)(2x) = f(2x) - g(2x) = (8x^3 + 1) - \frac{1}{2x}
\]

\[
= 8x^3 + 1 - \frac{1}{2x}
\]

### (c) \((f \circ g)(x)\)

\((f \circ g)(x) = f(g(x))\):

\[
f(g(x)) = f\left(\frac{1}{x}\right) = \left(\frac{1}{x}\right)^3 + 1
\]

\[
= \frac{1}{x^3} + 1
\]

### (d) \((g \circ f)(x)\)

\((g \circ f)(x) = g(f(x))\):

\[
g(f(x)) = g(x^3 + 1) = \frac{1}{x^3 + 1}
\]

### (e) \(((f - g) \circ g)(x)\)

\(((f - g) \circ g)(x) = (f - g)(g(x))\):

1. Find \( f(g(x)) \) and \( g(g(x)) \):

\[
f(g(x)) = f\left(\frac{1}{x}\right) = \frac{1}{x^3} + 1
\]

\[
g(g(x)) = g\left(\frac{1}{x}\right) = x
\]

2. Now calculate:

\[
(f - g)(g(x)) = f(g(x)) - g(g(x))
\]

\[
= \left(\frac{1}{x^3} + 1\right) - x
\]

\[
= \frac{1}{x^3} + 1 - x
\]

Would you like more details or have any questions?

Here are some related questions to expand on this topic:

1. How do you find the domain of a composite function?
2. What happens to the range of a function when it's composed with another function?
3. How can you determine if a composite function is invertible?
4

Tentukan nilai berikut, jika f(x) = x^3 + 1 dan g(x) = 1/x untuk (a) (f + g)(x) (b) (f − g)(2x) (c) (f ◦ g)(x) (d) (g ◦ f)(x) (e) ((f − g) ◦ g)(x)

Explore how to perform operations and compositions on functions with a detailed breakdown. Given f(x) = x^3 + 1 and g(x) = 1/x, learn to calculate (f + g)(x), (f - g)(2x), (f ◦ g)(x), and more. Ideal for high school students in Grades 10-12.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation