Problem 30 - College Algebra and Calculus: An Applied Approach

Question

Fill in the blank and simplify. \(f(t)=\sqrt{25-t^{2}}\) (a) \(f(3)=\sqrt{25-(})^{2}\) (b) \(f(5)=\sqrt{25-(\quad)^{2}}\) (c) \(f(x+5)=\sqrt{25-(\quad)^{2}}\) (d) \(f(2 x)=\sqrt{25-(\quad)^{2}}\)

Step-by-Step Solution

Answer

\(f(3)=4\), \(f(5)=0\), \(f(x+5)=\sqrt{25-(x+5)^{2}}\), \(f(2x)=\sqrt{25-4x^{2}}\)

Step 1: Substituting 3 into the function

We have to substitute 3 into the function, which gives: \(f(3)=\sqrt{25-3^{2}}\). Simplify this expression.

Step 2: Substituting 5 into the function

We have to substitute 5 into the function, which gives: \(f(5)=\sqrt{25-5^{2}}\). Simplify this expression.

Step 3: Substituting \(x+5\) into the function

We have to substitute \(x+5\) into the function, which gives: \(f(x+5)=\sqrt{25-(x+5)^{2}}\). This might not simplify a lot, but make sure to expand the squared term.

Step 4: Substituting \(2x\) into the function

We have to substitute \(2x\) into the function, which gives: \(f(2x)=\sqrt{25-(2x)^{2}}\). Here, make sure to correctly square the \(2x\) term.

Key Concepts

Substituting Variables

When it comes to solving mathematical problems, effectively substituting variables is essential. This process involves replacing the variable with a given number or another expression to evaluate a function or simplify an expression. In our example with the radical function f(t) = \(\sqrt{25-t^{2}}\), substituting variables like 3, 5, or expressions such as x+5 and 2x, transforms the abstract formula into specific values or new, perhaps more complex expressions.

However, care must be taken when substituting to ensure that the operations are correctly applied. For example, when substituting x+5 into the function, it's important to remember not only to replace t with x+5 but also to apply the square operation as in (x+5)^2.

Function Evaluation

The process of function evaluation is essentially computing the output of a function for a particular input. This is a foundational concept in algebra and higher mathematics. When you replace the variable in a function with a specific value (as you did with substituting variables), the next step is to perform the necessary operations to find the result, or the value of the function at that point.

For the radical function in our exercise, evaluating it at different points implies performing the square root operation after the substitution has been completed. If you substitute correctly and simplify, assessing the function’s value becomes much more straightforward.

Simplifying Expressions

The goal of simplifying expressions is to rewrite an expression in a simpler or more understandable form without changing its value. Simplification may involve combining like terms, reducing fractions, or utilizing properties of exponents and square roots.

In our exercise, after we substitute a value into the radical function, simplification could include squaring the number or expression that has been substituted and then determining the square root of the resulting value, provided it can be simplified. For instance, simplifying \(\sqrt{25-3^{2}}\) would involve calculating \(3^2\) to get \(9\), subtracting from \(25\) to get \(16\), and then finally deriving the square root of \(16\) which simplifies to \(4\).

Square Root Simplification

When dealing with radicals, particularly square root simplification, one must understand how to properly simplify square root expressions. This involves recognizing perfect squares and applying the square root to them to simplify the expression to its most basic form.

In expressions like \(\sqrt{25-t^{2}}\), after substituting and squaring the variable or number, you should look for a resulting perfect square under the radical. As we saw in the example above, \(\sqrt{25-9}}\) simplifies to \(\sqrt{16}\), and since \(16\) is a perfect square of \(4\), the expression simplifies further to just \(4\). Being able to recognize these patterns can significantly expedite the process of simplification.