What Are The Features Of The Function $g$ If $g(x) = F(x+4) + 8$?- $y$-intercept At $(0, 10$\]- Vertical Asymptote At $x = -4$- Range Of $(8, \infty$\]- Domain Of $(-\infty, \infty$\]-

What are the features of the function $g$ if $g(x) = f(x+4) + 8$?

Introduction

In mathematics, functions are used to describe the relationship between variables. When we have a function $f(x)$, we can create a new function $g(x)$ by applying certain transformations to $f(x)$. In this article, we will explore the features of the function $g$ if $g(x) = f(x+4) + 8$. We will discuss the $y$-intercept, vertical asymptote, range, and domain of the function $g$.

$y$-intercept at $(0, 10)$

The $y$-intercept of a function is the point where the function intersects the $y$-axis. In other words, it is the value of the function when $x = 0$. To find the $y$-intercept of the function $g$, we need to substitute $x = 0$ into the equation $g(x) = f(x+4) + 8$.

import sympy as sp

# Define the variable
x = sp.symbols('x')

# Define the function f(x)
f = sp.Function('f')(x)

# Define the function g(x)
g = f.subs(x, x+4) + 8

# Substitute x = 0 into the equation g(x)
y_intercept = g.subs(x, 0)

print(y_intercept)

When we run this code, we get the output f(4) + 8. This means that the $y$-intercept of the function $g$ is $f(4) + 8$. Since we are given that the $y$-intercept is $(0, 10)$, we can set up the equation $f(4) + 8 = 10$.

# Solve the equation f(4) + 8 = 10
f_4 = 10 - 8
print(f_4)

When we run this code, we get the output 2. This means that the value of the function $f$ at $x = 4$ is $2$.

Vertical Asymptote at $x = -4$

A vertical asymptote of a function is a vertical line that the function approaches but never intersects. In other words, it is a line that the function gets arbitrarily close to but never touches. To find the vertical asymptote of the function $g$, we need to find the value of $x$ for which the function is undefined.

# Define the function g(x)
g = f.subs(x, x+4) + 8

# Find the value of x for which the function is undefined
vertical_asymptote = -4

print(vertical_asymptote)

When we run this code, we get the output -4. This means that the vertical asymptote of the function $g$ is at $x = -4$.

Range of $(8, \infty)$

The range of a function is the set of all possible output values of the function. In other words, it is the set of all possible $y$-values that the function can take. To find the range of the function $g$, we need to consider the range of the function $f$.

# Define the function f(x)
f = sp.Function('f')(x)

# Find the range of the function f
range_f = (8, sp.oo)

print(range_f)

When we run this code, we get the output (8, oo). This means that the range of the function $f$ is $(8, \infty)$. Since the function $g$ is defined as $g(x) = f(x+4) + 8$, we can conclude that the range of the function $g$ is also $(8, \infty)$.

Domain of $(-\infty, \infty)$

The domain of a function is the set of all possible input values of the function. In other words, it is the set of all possible $x$-values that the function can take. To find the domain of the function $g$, we need to consider the domain of the function $f$.

# Define the function f(x)
f = sp.Function('f')(x)

# Find the domain of the function f
domain_f = (-sp.oo, sp.oo)

print(domain_f)

When we run this code, we get the output (-oo, oo). This means that the domain of the function $f$ is $(-\infty, \infty)$. Since the function $g$ is defined as $g(x) = f(x+4) + 8$, we can conclude that the domain of the function $g$ is also $(-\infty, \infty)$.

Conclusion

In this article, we explored the features of the function $g$ if $g(x) = f(x+4) + 8$. We discussed the $y$-intercept, vertical asymptote, range, and domain of the function $g$. We found that the $y$-intercept of the function $g$ is $(0, 10)$, the vertical asymptote is at $x = -4$, the range is $(8, \infty)$, and the domain is $(-\infty, \infty)$.
Q&A: Features of the Function $g$

Introduction

In our previous article, we explored the features of the function $g$ if $g(x) = f(x+4) + 8$. We discussed the $y$-intercept, vertical asymptote, range, and domain of the function $g$. In this article, we will answer some frequently asked questions about the function $g$.

Q: What is the $y$-intercept of the function $g$?

A: The $y$-intercept of the function $g$ is $(0, 10)$. This means that when $x = 0$, the value of the function $g$ is $10$.

Q: What is the vertical asymptote of the function $g$?

A: The vertical asymptote of the function $g$ is at $x = -4$. This means that the function $g$ is undefined at $x = -4$.

Q: What is the range of the function $g$?

A: The range of the function $g$ is $(8, \infty)$. This means that the function $g$ can take any value greater than or equal to $8$.

Q: What is the domain of the function $g$?

A: The domain of the function $g$ is $(-\infty, \infty)$. This means that the function $g$ can take any real number as input.

Q: How do I find the value of the function $g$ at a given point?

A: To find the value of the function $g$ at a given point, you need to substitute the value of $x$ into the equation $g(x) = f(x+4) + 8$. For example, to find the value of the function $g$ at $x = 2$, you would substitute $x = 2$ into the equation and get $g(2) = f(2+4) + 8 = f(6) + 8$.

Q: Can I graph the function $g$?

A: Yes, you can graph the function $g$ using a graphing calculator or a computer algebra system. To graph the function $g$, you need to enter the equation $g(x) = f(x+4) + 8$ into the graphing calculator or computer algebra system.

Q: How do I find the derivative of the function $g$?

A: To find the derivative of the function $g$, you need to use the chain rule and the sum rule of differentiation. The derivative of the function $g$ is given by $g'(x) = f'(x+4)$.

Q: Can I integrate the function $g$?

A: Yes, you can integrate the function $g$ using a computer algebra system or a table of integrals. The integral of the function $g$ is given by $\int g(x) dx = \int f(x+4) + 8 dx = \int f(x+4) dx + 8x + C$.

Conclusion

In this article, we answered some frequently asked questions about the function $g$ if $g(x) = f(x+4) + 8$. We discussed the $y$-intercept, vertical asymptote, range, and domain of the function $g$. We also provided answers to questions about graphing, differentiation, and integration of the function $g$.