Memristive crossbar arrays promise substantial improvements in computing throughput and power efficiency through in-memory analog computing. Previous machine learning demonstrations with memristive arrays, however, relied on software or digital processors to implement some critical functionalities, leading to frequent analog/digital conversions and more complicated hardware that compromises the energy efficiency and computing parallelism. Here, we show that, by implementing the activation function of a neural network in analog hardware, analog signals can be transmitted to the next layer without unnecessary digital conversion, communication, and processing. We have designed and built compact rectified linear units, with which we constructed a two-layer perceptron using memristive crossbar arrays, and demonstrated a recognition accuracy of 93.63% for the Modified National Institute of Standard and Technology (MNIST) handwritten digits dataset. The fully hardware-based neural network reduces both the data shuttling and conversion, capable of delivering much higher computing throughput and power efficiency.