Question

which function has an inverse that is a function?

\\(b(x) = x^2 + 3\\)
\\(d(x) = -9\\)
\\(m(x) = -7x\\)
\\(p(x) = |x|\\)

Explanation:

Identify the condition for an invertible function

Using the One-to-One Functions knowledge point
A function has an inverse that is also a function if and only if the original function is one-to-one, meaning each output value corresponds to exactly one input value.

Apply the horizontal line test

Using the Horizontal Line Test knowledge point
A function is one-to-one if and only if no horizontal line intersects its graph more than once.

Analyze the given options

Using the One-to-One Functions and Horizontal Line Test knowledge points

• For \(b(x) = x^2 + 3\), the horizontal line \(y = 4\) intersects the graph at \(x = 1\) and \(x = -1\). It is not one-to-one.
• For \(d(x) = -9\), the horizontal line \(y = -9\) intersects the graph at infinitely many points. It is not one-to-one.
• For \(m(x) = -7x\), any horizontal line \(y = c\) intersects the graph at exactly one point, \(x = -\frac{c}{7}\). It is one-to-one.
• For \(p(x) = |x|\), the horizontal line \(y = 1\) intersects the graph at \(x = 1\) and \(x = -1\). It is not one-to-one.

Select the correct function

Using the One-to-One Functions knowledge point
The only function that is one-to-one, and therefore has an inverse that is a function, is \(m(x) = -7x\).

Answer:

• (A) \(b(x) = x^2 + 3\)
• (B) \(d(x) = -9\)
• (C) \(m(x) = -7x\) (Correct answer)
• (D) \(p(x) = |x|\)