Problem 15 - Calculus: An Integrated Approach to Functions and Their Rates of Change

Question

If \(g(x)=\frac{\sqrt{x^{2}+4}}{2}\), find the following. Simplify your answer where possible. (a) \(g(0)\) (b) \(g(2)\) (c) \(g(\sqrt{5})\) (d) \(g\left(\frac{1}{\sqrt{2}}\right)\) (e) \(-g(3 t)\) (f) \(g(\sqrt{t-4})\)

Step-by-Step Solution

Answer

(a) \(g(0)=1\), (b) \(g(2)=\sqrt{2}\), (c) \(g(\sqrt{5})=\frac{3}{2}\), (d) \(g(\frac{1}{\sqrt{2}})=\frac{3}{2\sqrt{2}}\), (e) \(-g(3 t) = – \frac{\sqrt{9t^{2}+4}}{2}\), (f) \(g(\sqrt{t-4})=\frac{\sqrt{t+4}}{2}\)

Step 1: Substitute and simplify for \(g(0)\)

First, substitute \(x=0\) into the function to get \(g(0)=\frac{\sqrt{0^{2}+4}}{2}=\frac{\sqrt{4}}{2}=1\)

Step 2: Substitute and simplify for \(g(2)\)

Next, substitute \(x=2\) into the function to get \(g(2)=\frac{\sqrt{2^{2}+4}}{2}=\frac{\sqrt{8}}{2}= \sqrt{2}\)

Step 3: Substitute and simplify for \(g(\sqrt{5})\)

For \(x=\sqrt{5}\), substitute it into the function to get \(g(\sqrt{5})=\frac{\sqrt{(\sqrt{5})^{2}+4}}{2}=\frac{\sqrt{5+4}}{2}=\frac{\sqrt{9}}{2}= \frac{3}{2}\)

Step 4: Substitute and simplify for \(g(\frac{1}{\sqrt{2}})\)

For \(x=\frac{1}{\sqrt{2}}\), substitute it into the function to get \(g(\frac{1}{\sqrt{2}})=\frac{\sqrt{(\frac{1}{\sqrt{2}})^{2}+4}}{2}=\frac{\sqrt{\frac{1}{2}+4}}{2}=\frac{\sqrt{\frac{9}{2}}}{2}=\frac{3}{2\sqrt{2}}\)

Step 5: Substitute and simplify for \(-g(3t)\)

For \(x=3t\), since there is a negative sign before \(g(3t)\), firstly substitute \(x=3t\) into the function to get \(g(3t)=\frac{\sqrt{(3t)^{2}+4}}{2}=\frac{\sqrt{9t^{2}+4}}{2}\), then it's negative value is \(-g(3t)= – \frac{\sqrt{9t^{2}+4}}{2}\)

Step 6: Substitute and simplify for \(g(\sqrt{t-4})\)

For \(x=\sqrt{t-4}\), substitute it into the function to get \(g(\sqrt{t-4})=\frac{\sqrt{(\sqrt{t-4})^{2}+4}}{2}=\frac{\sqrt{t+4}}{2}\)

Key Concepts

Simplifying Expressions

Simplifying expressions is an essential concept in mathematics, one that simplifies complex structures into simpler and more manageable forms. This process allows us to handle challenging parts of expressions easily and efficiently.
When dealing with functions, as seen in the step-by-step solution, simplifying is crucial to make substitution and evaluation more straightforward.

• Start by looking for common elements or factors.
• Combine like terms where possible.
• Resolve any arithmetic operations, such as addition or multiplication, inside parentheses.

For example, when you have an expression like \[\frac{\sqrt{x^2+4}}{2}\], identifying terms like \(x^2 + 4\) allows for easy evaluation once you substitute values into \(x\). This is also helpful in identifying opportunities to reduce fractions or expressions, as seen in cases like \(\sqrt{9}\) becoming \(3\). With expressions that involve square roots, simplification paves the way for clearer understanding and computation.

Square Roots

Square roots are among the most common mathematical operations encountered during function evaluations. Understanding square roots means you know how to find a value that, when multiplied by itself, gives the original number. This principle underpins much of the algebraic manipulation seen when solving or simplifying functions.
In the given function, \(g(x)=\frac{\sqrt{x^2+4}}{2}\), the square root \(\sqrt{x^2+4}\) is central. By computing the square root first, you simplify the inner workings of the function and prepare it for further calculation or simplification.

• To work with square roots, recognize them as fractional exponents \((x^{1/2})\).
• When simplifying square root expressions such as \(\sqrt{8}\), which appear in \(g(2)\), you can express it as \(2\sqrt{2}\) for simplifying calculations.
• This transformation helps in making expressions tidy, allowing further evaluations to proceed smoothly.

Square roots can often result in simple integer values, as highlighted in \(\sqrt{9}\) for \(g(\sqrt{5})\) resulting in \(\frac{3}{2}\), contributing to easy computation.

Substitution in Functions

Substitution is a technique used to evaluate a function at a specific point. It involves replacing the variable in the function with a given value, allowing us to transform an abstract expression into a numeric one. With functions, this is a straightforward process that provides concrete results in real-world applications or problem-solving.
The function \(g(x)=\frac{\sqrt{x^2+4}}{2}\) is a great example of using substitution for evaluation.

• Start by identifying the function's rule, then replace the variable \(x\) with the given number.
• For example, by substituting \(x=0\), \(x=2\), or \(x=\sqrt{5}\) into \(g(x)\), we simplify the expressions to numeric outputs.
• This shift simplifies complex functions into understandable and manageable numbers.

Moreover, substitutions can involve expressions rather than just numbers, shown by \(g(\frac{1}{\sqrt{2}})\), highlighting the flexibility and power of substitution in functions.