Question

Are based on material learned earlier in the course. The purpose of these problems is to keep the material fresh in your mind so that you are better prepared for the final exam. If \(f(x)=\sqrt{3-5 x}\) and \(g(x)=x^{2}+7,\) find \(g(f(x))\) and its domain.

Short answer

The composite function is \(g(f(x)) = 10 - 5x\) with domain \((-\infty, \frac{3}{5}]\).

Step-by-step solution

- Understand the problem

We need to find the composite function \(g(f(x))\) where \(f(x) = \sqrt{3 - 5x}\) and \(g(x) = x^2 + 7\).

- Define the inner function

\(f(x) = \sqrt{3 - 5x}\). We start by finding this expression inside the function \(g\).

- Substitute \(f(x)\) into \(g(x)\)

Substitute \(f(x)\) into \(g(x)\) to get \(g(f(x)) = g(\sqrt{3 - 5x})\). Since \(g(x) = x^2 + 7\), replace \(x\) with \(\sqrt{3 - 5x}\). Therefore, \(g(f(x)) = (\sqrt{3 - 5x})^2 + 7\).

- Simplify the expression

Simplify \((\sqrt{3 - 5x})^2 + 7\) to get \(3 - 5x + 7\). Thus, \(g(f(x)) = 10 - 5x\).

- Find the domain of \(f(x)\)

The domain of \(f(x)\) is determined by the requirement that the expression inside the square root must be non-negative. So, \(3 - 5x \geq 0\). Solve for \(x\): \(3 \geq 5x\) \(x \leq \frac{3}{5}\). Therefore, the domain of \(f(x)\) is \((-\infty, \frac{3}{5}]\).

- Determine the domain of \(g(f(x))\)

Since \(g(f(x)) = 10 - 5x\), the domain of \(g(f(x))\) is the same as the domain of \(f(x)\), which is \((-\infty, \frac{3}{5}]\).

Key concepts

Function Composition

Function composition means combining two functions to create a new function. Think of it as applying one function to the results of another. This makes a new function based on the formulas of the two original ones.
For example, if we have two functions, f(x) and g(x), the composite function g(f(x)) means we first apply f to x, and then g to the result of f(x).
Why do we use function composition? It's useful for simplifying complex problems by breaking them down into simpler tasks.
In our exercise, we start with two given functions: f(x) = \(\sqrt{3-5x}\) and g(x) = \(x^2 + 7\). To find g(f(x)), we follow these steps:

• Identify the inner function f(x) = \(\sqrt{3-5x}\).
• Substitute f(x) into g. This means wherever there's an x in g(x), we replace it with f(x).
• Simplify the resulting expression to find the final composite function.

Domain of Functions

The domain of a function is the set of all possible input values (x-values) that the function can accept without causing any mathematical errors.
For functions involving a square root, like f(x) = \(\sqrt{3 - 5x}\), the radicand (the expression inside the square root) must be non-negative. This is because the square root of a negative number is not a real number.
To find the domain of f(x):

• Set the expression inside the square root to be greater than or equal to 0: 3 - 5x ≥ 0.
• Solve the inequality for x to find the possible values: 3 ≥ 5x, which simplifies to \(x \leq \frac{3}{5}\).

So, the domain of f(x) is \((-\infty, \frac{3}{5}]\).
For the composite function g(f(x)), the domain is also impacted by the domain of f(x). Since g depends on the output of f, the same limitations on x-values apply.
Therefore, the domain of g(f(x)) = 10 - 5x is also \((-\infty, \frac{3}{5}]\).

Simplifying Expressions

Simplifying expressions in function composition helps us make them more manageable and understandable.
In our problem, we substitute f(x) into g(x) and simplify:

• Substitute: g(f(x)) = g(\(\sqrt{3 - 5x}\))
• Since g(x) = \(x^2 + 7\), we replace x with \(\sqrt{3 - 5x}\).
• This gives us: g(f(x)) = (\(\sqrt{3 - 5x}\))^2 + 7.
• Simplify the expression: (\(\sqrt{3 - 5x}\))^2 = 3 - 5x.
• So, g(f(x)) = 3 - 5x + 7.
• Combine like terms to finalize: g(f(x)) = 10 - 5x.

Simplifying makes it clear that our final composite function g(f(x)) is a linear function, which is easier to work with.
Always remember when simplifying:

• Carry out operations like squaring, adding, or subtracting carefully.
• Combine like terms to make the expression as simple as possible.