Question

the graph of a parent function is shown.

which function belongs to the same function family as the graphed function?

\\(f(x) = 5^x - 4\\)
\\(f(x) = \frac{1}{5}\sqrt{x + 4}\\)
\\(f(x) = 5x^2 + 4\\)
\\(f(x) = 5\sqrt{x} - 4\\)

Explanation:

Identify the parent function family from the graph

The graph starts at the origin \((0,0)\) and curves gradually to the right in the first quadrant. This shape represents the parent square root function, \(y = \sqrt{x}\).

Analyze the given function options

• \(f(x) = 5^x - 4\) is an exponential function.
• \(f(x) = \frac{1}{5}\sqrt{x} + 4\) is a transformed square root function.
• \(f(x) = 5x^2 + 4\) is a quadratic function.
• \(f(x) = 5\sqrt{x} - 4\) is a transformed square root function.

Select the matching option from the visible choices

Looking closely at the options in the image:

• Option 1: \(f(x) = 5^x - 4\)
• Option 2: \(f(x) = \frac{1}{5}\sqrt{x+4}\)
• Option 3: \(f(x) = 5x^2 + 4\)
• Option 4: \(f(x) = 5\sqrt{x} - 4\)

Both Option 2 and Option 4 belong to the square root function family. However, Option 4, \(f(x) = 5\sqrt{x} - 4\), is the standard representation of a transformed square root function \(a\sqrt{x} + k\).

Answer:

• (A) \(f(x) = 5^x - 4\)
• (B) \(f(x) = \frac{1}{5}\sqrt{x+4}\)
• (C) \(f(x) = 5x^2 + 4\)
• (D) \(f(x) = 5\sqrt{x} - 4\) (Correct answer)