Question 25

f(x) = -4x² + 5 has more than one x-intercept.

The function f(x) = -4x² + 5 is a quadratic function that opens downwards, indicating it can intersect the x-axis at two points. This characteristic leads to the possibility of having two real solutions for the equation -4x² + 5 = 0, confirming that it has more than one x-intercept.

A) f(x) = -4x - 5

This function is a linear equation with a negative slope, which means it crosses the x-axis at exactly one point. Therefore, it does not have more than one x-intercept.

B) f(x) = -4x² + 5

As a quadratic function, it has the potential to intersect the x-axis at two points, particularly since its leading coefficient is negative. The discriminant of this quadratic is positive (16), confirming that it indeed has two real x-intercepts.

C) f(x) = 4

This is a constant function, represented by a horizontal line at y = 4. It does not intersect the x-axis at all, resulting in zero x-intercepts.

D) f(x) = 4x² + 5

This function is also a quadratic, but it opens upwards with a positive leading coefficient. Since its vertex does not reach the x-axis, it has no real x-intercepts and therefore cannot have more than one.

E) f(x) = 4x^3 + 5

This is a cubic function that may have one or three x-intercepts, but it does not guarantee more than one x-intercept. In this specific case, it has only one real root, thus not fulfilling the requirement of having more than one x-intercept.

Conclusion

The function f(x) = -4x² + 5 is the only one among the options that is guaranteed to have more than one x-intercept due to its downward-facing parabola. All other functions either have one or no x-intercepts, making option B the definitive choice for this question.