Question

For the curve \(y=x^{3}+1\), the values of \(y\) corresponding to values of \(x\) as \(-2\) and 2 are (a) \(7,-9\) (b) \(-3,6\) (c) \(-7,9\) (d) \(6,-3\)

Short answer

So the values of \(y\) corresponding to \(x = -2\) and \(x = 2\) are \(-7\) and \(9\) respectively. Therefore, the solution is (c) \(-7,9\).

Step-by-step solution

Substitute \(x=-2\)

Substitute \(x = -2\) into the equation. That means calculating \((-2)^{3}+1 = -8 + 1 = -7\). Therefore, when \(x = -2\), \(y = -7\).

Substitute \(x=2\)

Now, substitute \(x = 2\) into the equation. That gives \(2^{3}+1 = 8 + 1 =9\). So, when \(x = 2\), \(y = 9\).

Key concepts

Algebra

Algebra is a branch of mathematics dealing with symbols and the rules for manipulating these symbols; it is a unifying thread of almost all of mathematics. It gives us the tools to describe and solve problems with unknown variables and complex relationships. In the context of cubic functions, algebra allows us to understand how equations represent curves and how these curves change as their variables change.

At its core, algebra involves solving equations and finding the unknowns. For the cubic function exercise given, algebra is used to work with the cubic function equation and execute function evaluations by substituting specific values for the variable 'x'. Understanding the basic operations like addition, subtraction, exponentiation, and applying them correctly is essential to solving algebraic equations in exercises such as these.

Function Evaluation

Function evaluation is the process of determining the output of a function for a particular input. This procedure is like following a recipe where the input value is the ingredients and the function itself is the recipe telling you what to do with these ingredients. In our cubic function example, the 'recipe' is to take whatever value of 'x' is given and raise it to the third power, then add 1 to that result to find 'y'.

Therefore, when you're given specific values of 'x', such as -2 or 2, you can plug them into the function—to replace the variable with the actual numbers—and perform the indicated operations: In this case, cube the number and add one. Function evaluation requires careful arithmetic to avoid simple mistakes that could lead to incorrect answers. It's fundamental in algebra to understand how to evaluate functions properly.

Solving Cubic Equations

Solving cubic equations can be a more complex process than solving linear or quadratic equations due to the higher exponent. Cubic equations have the form \( ax^3 + bx^2 + cx + d = 0 \), where 'a', 'b', 'c', and 'd' are constants, and 'x' is the variable. These equations can sometimes be solved by factoring, though more often require more complex methods such as synthetic division or the use of the cubic formula.

In the exercise given, we are not directly solving a cubic equation but evaluating the function \( y = x^3 + 1 \) for given 'x' values. However, understanding the concept of solving cubic equations is beneficial for when you encounter equations where you're required to find 'x' when 'y' is given. In those cases, a methodical approach to solving might be needed, including, in some cases, graphing the function to visually identify the roots or points of intersection with the 'x-axis'.