Question

In the form \(y=\) \(k \cdot(h(x))^{p}\) for some function \(h(x)\). $$ y=\frac{17}{\left(1-x^{3}\right)^{4}} $$

Short answer

Question: Express the function \(y=\frac{17}{\left(1-x^{3}\right)^{4}}\) in the form \(y=k\cdot(h(x))^{p}\). Answer: The function can be expressed as \(y=17 \cdot (1-x^3)^{-4}\).

Step-by-step solution

Identify the Constant Factor in the Given Function

We first need to find the constant factor \(k\). In the given function, the constant factor is \(17\).

Identify the Power in the Given Function

Now, we need to find the value of \(p\). The power in the given function is the exponent on the denominator term, which is \(4\). Since the exponent is in the denominator and we want to express the function in the form \(y=k\cdot(h(x))^{p}\), we should use a negative exponent, so \(p=-4\).

Identify the Function h(x) in the Given Function

Lastly, we need to find the function \(h(x)\). This function can be found in the denominator of the given function, which is \((1-x^3)\). As we want to express the given function in terms of \(k,h(x),p\), we have to bring the denominator to the numerator using a negative exponent (as we already found out that \(p=-4\) in Step 2). So, the function \(h(x)\) is \((1-x^3)\).

Rewrite the Given Function in the Required Form

Now that we have \(k=17\), \(p=-4\), and \(h(x)=(1-x^3)\), we can rewrite the given function in the form \(y=k\cdot(h(x))^p\): $$ y=17 \cdot (1-x^3)^{-4} $$ This is the given function expressed in the form \(y=k\cdot(h(x))^{p}\).

Key concepts

Exponents

When dealing with rational functions, exponents play a vital role. They're the numbers that tell you how many times to multiply a base number, which can drastically change the value of an expression.
In our task, we have an exponent in a denominator: - The expression is \(\left(1-x^{3}\right)^4\).- When we bring this term to the numerator, it becomes \(\left(1-x^{3}\right)^{-4}\).Understanding exponents involves knowing that:- A positive exponent, such as \(n\), indicates multiplication \(n\) times.- A negative exponent, like \(-n\), implies division or taking the reciprocal. Hence, \(a^{-n}\) means \(\frac{1}{a^n}\).This is how we shift from the denominator to the numerator in our function transformation. This concept helps simplify expressions and solve equations efficiently.

Function Transformation

Function transformation refers to the modifications we make to a function's basic form. Transformations can scale, shift, or reflect a function based on specific operations.
In the transformation of the given expression, - The original is \(\frac{17}{(1-x^3)^4}\), - We transform it into \(17 \cdot (1-x^3)^{-4}\).The process includes:- Identifying the constant factor, which in this case is 17.- Moving the exponent from the denominator to the numerator by changing it from positive to negative (reflecting across the axis).Function transformation helps in analyzing and sketching functions easily. By understanding these small adjustments, you gain a deeper insight into the shape and behavior of graphs across different mathematical contexts.

Algebraic Expressions

An algebraic expression is a combination of numbers, variables, and operators such as addition, subtraction, multiplication, and division. Rational functions like our given problem are algebraic expressions because they involve variables and exponents.
The primary components here include:- **Variables**: Elements like \(x\) represent unknown values that can change.
- **Constants**: Here, 17 is a constant unchanging value.
- **Operators and Exponents**: Combine and manipulate these constants and variables. When evaluating algebraic expressions:- Identify constants and variables.
- Recognize operators’ roles and sequence of operations.- Understand how exponents affect terms.Rewriting rational functions as algebraic expressions aids in their easy manipulation and understanding, allowing for simplified expressions or solutions in calculus and advanced algebra.