Problem 36 - College Algebra with Corequisite Support

Question

If possible, find all values of \(a\) such that there are no \(x\) - intercepts for \(f(x)=2|x+1|+a\).

Step-by-Step Solution

Answer

The values of \( a \) must be positive.

Step 1: Understanding x-intercepts

First, let's understand what it means for a function to have no x-intercepts. An x-intercept occurs when the function is equal to zero at some point, i.e., when \( f(x) = 0 \). For \( f(x) = 2|x+1|+a \), there will be no x-intercepts if the equation \( 2|x+1| + a = 0 \) has no solution for \( x \).

Step 2: Setting up the equation

To find where the x-intercepts would occur, set \( f(x) = 0 \). This gives the equation \( 2|x+1| + a = 0 \). Rearrange this equation to isolate the absolute value expression: \( 2|x+1| = -a \).

Step 3: Analyzing the absolute value condition

Recall that an absolute value expression, \( |x+1| \), is always non-negative (i.e., \( \geq 0 \)). Therefore, \( 2|x+1| \geq 0 \) for all \( x \). For \( 2|x+1| = -a \) to hold with no solutions, \( -a \) must be negative. Thus, \( a \) must be positive, since \( -a < 0 \) when \( a > 0 \).

Step 4: Conclusion on values of a

The equation \( 2|x+1| = -a \) can have no solutions only if \( a > 0 \). Therefore, for the function \( f(x) = 2|x+1| + a \) to have no x-intercepts, the value of \( a \) must be any positive number.

Key Concepts

x-intercepts

In algebra, x-intercepts are points where a function crosses the x-axis. This means the function's value is zero, so finding x-intercepts involves solving the equation where the function equals zero. For example, if we have a function like \( f(x) = 2|x+1| + a \), to find its x-intercepts, we must solve \( f(x) = 0 \). X-intercepts impose valuable insights into a function's graph, indicating where the function changes its sign across the x-axis.

absolute value

The absolute value of a number signifies its distance from zero on the number line, regardless of direction. Absolute value functions, denoted by \(|x|\), take any real number \(x\) and convert it to a non-negative value. This means \(|x|\) equals \(x\) if \(x\) is positive or zero, and \(-x\) if \(x\) is negative. In the function \(f(x) = 2|x+1| + a\), the expression \(|x+1|\) affects how the function behaves around \(-1\). Because absolute values yield non-negative results, they inherently influence the solutions when solving equations involving them.

functions

Functions in mathematics describe a relationship between a set of inputs and a corresponding set of outputs. In other words, for every input value, there is exactly one output. Functions like \( f(x) = 2|x+1| + a \) are vital tools in algebra because they help model real-world phenomena and solve various problems. Understanding how a function operates and transforms based on input values is central to analyzing and predicting behaviors, especially when determining intercepts, maxima, minima, or critical points.

solving equations

Solving equations involves finding values that satisfy an equation. There are various techniques depending on the type and complexity of the equation. When dealing with the equation \( 2|x+1| + a = 0 \), we want to isolate the variable to find potential solutions. For an equation like this, it's crucial to consider conditions like the non-negativity of absolute values, which restricts possible solutions. Often, this process involves rearranging terms and understanding implied mathematical properties to determine valid solutions or identify when no solutions exist. Solving such equations often combines algebraic manipulations with logical reasoning to reach a solution or verify its absence.