Question

Find the domain of \(h(x)=\sqrt[4]{x+7}+7 x\).

Short answer

The domain of \(h(x) = \sqrt[4]{x+7} + 7x\) is \(x \geq -7\).

Step-by-step solution

- Understand the Function Components

The function given is composed of two parts: the fourth root function and a linear function. The first part is \(\sqrt[4]{x+7}\), and the second part is \(7x\). Understand how each part affects the domain.

- Determine the Domain of the Fourth Root Function

The fourth root function \(\sqrt[4]{x+7}\) is defined for values where the expression under the root is non-negative. So, set up the inequality \(x + 7 \geq 0\). Solving this inequality gives \(x \geq -7\).

- Identify the Domain of the Linear Function

The linear function \(7x\) is defined for all real numbers, meaning it does not restrict the domain further.

- Combine the Results

Since the linear function does not add any new restrictions to the domain, the domain is determined solely by the fourth root function. Therefore, the combined domain of \(h(x)\) is all values \(x \geq -7\).

Key concepts

domain determination

Determining the domain of a function involves finding all possible values of the independent variable (often denoted as 'x') for which the function is defined. The domain is crucial because it tells us where the function can operate.

In the given problem, we aim to find the domain of the function \(h(x)=\sqrt[4]{x+7}+7x\). This requires an understanding of both the fourth root function \(\sqrt[4]{x+7}\) and the linear function \(7x\).

To start, we analyze each part of the function separately to determine their individual domains. Afterward, we combine these domains to find the overall domain of the function.

fourth root function

The fourth root function, denoted as \(\sqrt[4]{x+7}\), is a special type of root function. It is defined for values where the expression under the root is non-negative.

In mathematical terms, the inequality \(x + 7 \geq 0\) must hold true for the fourth root to be valid. To solve this inequality:

• Subtract 7 from both sides to isolate x: \(x \geq -7\).

This tells us that for the fourth root function to be real and defined, x must be greater than or equal to -7. Therefore, the domain of \(\sqrt[4]{x+7}\) is all values \(x \geq -7\).

solving inequalities

Solving inequalities such as \(x + 7 \geq 0\) is a method to determine valid input values for functions. Here's a quick guide:

• First, isolate the variable (x) by performing inverse operations on both sides of the inequality.
• For the given example: \(x + 7 \geq 0\). Subtract 7 from both sides to get \(x \geq -7\).
• This means x must be -7 or greater to satisfy the inequality.

Understanding the result of solving this inequality directly impacts the function's domain.

linear function properties

Linear functions, such as \(7x\), exhibit simple and well-defined properties. They are polynomials of degree one and are defined for all real numbers.

The domain of a linear function is typically all real numbers, \( (-\infty, \infty) \). Here, the function \(7x\) does not impose any additional restrictions on its input values. It means any real number works for the linear part of our combined function.

Therefore, the domain constraint will primarily come from the non-linear part of the function.

combined domain

After determining the domains of individual components, we need to combine them. The function \(h(x)=\sqrt[4]{x+7}+7x\) has two parts:

• The fourth root function, which requires \(x \geq -7\).
• The linear function, which allows all real numbers.

Since these parts work together in the given function, the overall domain is dictated by the more restrictive part.

Thus, the combined domain for \(h(x)\) is \(x \geq -7\). We ensure the function is valid by confirming any x-value meet this requirement, consolidating both components seamlessly.