Question

column 1 | f(x) | g(x) | both f(x) and g(x) | neither f(x) and g(x)
the domain is -∞ < x < ∞ | ○ | ○ | ○ | ○
the range is y ≥ -2 | ○ | ○ | ○ | ○
an x - intercept of the function is -1 | ○ | ○ | ○ | ○
the y - intercept of the function is 3 | ○ | ○ | ○ | ○

Explanation:

To solve this, we need to analyze the domain, range, x - intercept, and y - intercept for functions \(f(x)\) and \(g(x)\). However, since the actual functions \(f(x)\) and \(g(x)\) are not provided (only the table to fill based on their properties), we assume that we have the graphs or equations of \(f(x)\) and \(g(x)\) (which are not shown here). Let's assume typical function properties:

Step 1: Domain \(-\infty

If a function is a polynomial (like linear, quadratic) or exponential, its domain is all real numbers. For example, a quadratic function \(y = ax^{2}+bx + c\) has domain \((-\infty,\infty)\). If both \(f(x)\) and \(g(x)\) are such functions, we would select "Both \(f(x)\) and \(g(x)\)".

Step 2: Range \(y\geq - 2\)

A parabola opening upwards with vertex at \(y=-2\) (like \(y=(x - h)^{2}-2\)) has range \(y\geq - 2\). If one of the functions is such a parabola and the other is not, we select the appropriate column.

Step 3: x - intercept at \(x = - 1\)

The x - intercept is where \(y = 0\). If \(f(-1)=0\) or \(g(-1)=0\), we select the corresponding column. For example, if \(f(x)=x + 1\), then \(f(-1)=0\), so we select \(f(x)\).

Step 4: y - intercept at \(y = 3\)

The y - intercept is where \(x = 0\). If \(f(0)=3\) or \(g(0)=3\), we select the corresponding column. For example, if \(f(x)=x + 3\), then \(f(0)=3\), so we select \(f(x)\).

Since the actual functions are not provided, we can't give a definite answer without more information about \(f(x)\) and \(g(x)\). But if we assume standard functions:

For example, if \(f(x)=(x + 1)^{2}-2=x^{2}+2x - 1\)

• Domain: \((-\infty,\infty)\) (so "Both" if \(g(x)\) also has all real numbers domain)
• Range: \(y\geq - 2\) (since the vertex is at \(y=-2\) and it opens upwards)
• x - intercept: Solve \(x^{2}+2x - 1=0\), \(x=-1\pm\sqrt{2}\), so x - intercept is not \(-1\) (wait, if \(f(x)=x + 1\), then x - intercept is \(-1\))
• y - intercept: \(f(0)=-1\), not 3.

If \(f(x)=x + 3\)

• Domain: \((-\infty,\infty)\)
• Range: \((-\infty,\infty)\) (not \(y\geq - 2\))
• x - intercept: \(x=-3\) (not \(-1\))
• y - intercept: \(y = 3\)

Answer:

Without the actual definitions of \(f(x)\) and \(g(x)\), we can't provide a definite answer. Please provide the equations or graphs of \(f(x)\) and \(g(x)\) to determine the correct selections.