To solve the equation \( x^3 - 5 = -x + 5 \) using function graphs, let's first rewrite the equation in a more manageable form:
- Rearrange the equation: \[ x^3 + x - 5 = 0 \]
Next, we can think of this as finding the x-values where the function \( f(x) = x^3 + x - 5 \) intersects the x-axis (where \( f(x) = 0 \)).
Using a graph to visualize \( f(x) = x^3 + x - 5 \), we observe the following:
- The function is a cubic polynomial, which generally has one or more points where it intersects the x-axis.
- By evaluating the function at specific x-values:
- \( f(0) = 0^3 + 0 - 5 = -5 \) (below x-axis)
- \( f(2) = 2^3 + 2 - 5 = 8 + 2 - 5 = 5 \) (above x-axis)
- \( f(1) = 1^3 + 1 - 5 = 1 + 1 - 5 = -3 \) (below x-axis)
- \( f(3) = 3^3 + 3 - 5 = 27 + 3 - 5 = 25 \) (above x-axis)
From this analysis:
- The x-axis is crossed between 1 and 2.
- It indicates that \( x = 2 \) is a solution since \( f(2) = 5 \) is not correct.
- On re-evaluating the given options, it shows \( x = 3 \) as a potential solution.
Thus, the solution from the given responses should be checked more. Evaluating again at needed critical points between 1 and 3, we find that \( x = 2 \) is relevant.
Thus, the correct answer is:
- x= 2